{"group":{"group":{"id":46175,"name":"Special Functions II","lockable":false,"created_at":"2022-06-06T02:05:43.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"More on special functions and their applications. Some are available in MATLAB, and some are not. ","is_default":false,"created_by":46909,"badge_id":62,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":3290,"published":true,"community_created":true,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMore on special functions and their applications. Some are available in MATLAB, and some are not. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 290px 21px; transform-origin: 290px 21px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 267px 21px; text-align: left; transform-origin: 267px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 265.558px 8px; transform-origin: 265.558px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMore on special functions and their applications. Some are available in MATLAB, and some are not. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","published_at":"2022-06-06T02:32:09.000Z"},"current_player":null},"problems":[{"id":45939,"title":"Estimate π from certain values of the zeta function","description":"Cody Problems \u003chttps://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi 2908\u003e and \u003chttps://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs 2909\u003e ask us to estimate π by summing a given number of terms in an infinite sum, the Leibniz formula.\r\n\r\nHere you are asked to estimate π by summing a given number of terms in other infinite sums corresponding to the \u003chttps://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function\u003e evaluated at positive even integers. For example, in solving the \u003chttps://en.wikipedia.org/wiki/Basel_problem Basel problem\u003e, Euler summed the reciprocals of the squares of positive integers and showed that ζ(2) = π^2/6.\r\n\r\nWrite a function that takes a vector n with the numbers of terms to sum and a vector m with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is, |abs(E-pi)/pi|. Report the relative error in scientific notation using num2str(...,'%10.2e').\r\n\r\nFor example, PiByZeta(30,2) should return\r\n\r\n '1.00e-02'\r\n \r\nwhile PiByZeta([25 30],[2 8]) should return \r\n\r\n '1.20e-02  1.00e-02'\r\n '2.53e-12  7.22e-13'\r\n  \r\n ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 329.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 164.6px; transform-origin: 407px 164.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.85px 7.8px; transform-origin: 47.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCody Problems\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e2908\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003e2909\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2833px 7.8px; transform-origin: 60.2833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ask us to estimate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.217px 7.8px; transform-origin: 211.217px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e by summing a given number of terms in an infinite sum, the Leibniz formula.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1833px 7.8px; transform-origin: 99.1833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHere you are asked to estimate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.95px 7.8px; transform-origin: 248.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e by summing a given number of terms in other infinite sums corresponding to the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Riemann_zeta_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"perspective-origin: 69.6333px 7.8px; transform-origin: 69.6333px 7.8px; \"\u003eRiemann zeta function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 197.217px 7.8px; transform-origin: 197.217px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e evaluated at positive even integers. For example, in solving the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Basel_problem\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eBasel problem\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6167px 7.8px; transform-origin: 62.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, Euler summed the reciprocals of the squares of positive integers and showed that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"zeta(2) = pi^2/6\" style=\"width: 78px; height: 19.5px;\" width=\"78\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.95px; text-align: left; transform-origin: 384px 31.95px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.55px 7.8px; transform-origin: 109.55px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.783px 7.8px; transform-origin: 147.783px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with the numbers of terms to sum and a vector \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 113.967px 7.8px; transform-origin: 113.967px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2px 7.8px; transform-origin: 46.2px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 46.2px 8.25px; transform-origin: 46.2px 8.25px; \"\u003eabs(E-pi)/pi\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 227.483px 7.8px; transform-origin: 227.483px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Report the relative error in scientific notation using num2str(...,'%10.2e').\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.25px 7.8px; transform-origin: 132.25px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, PiByZeta(30,2) should return\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20.4333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 42.35px 8.25px; transform-origin: 42.35px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; \"\u003e \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 38.5px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 38.5px 8.25px; \"\u003e'1.00e-02'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.7px 7.8px; transform-origin: 130.7px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhile PiByZeta([25 30],[2 8]) should return\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40.8667px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 20.4333px; transform-origin: 404px 20.4333px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80.85px 8.25px; transform-origin: 80.85px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; \"\u003e \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 77px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 77px 8.25px; \"\u003e'1.20e-02  1.00e-02'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80.85px 8.25px; transform-origin: 80.85px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; \"\u003e \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 77px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 77px 8.25px; \"\u003e'2.53e-12  7.22e-13'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = PiByZeta(n,m)\r\n  s = num2str(ans,'%10.2e');\r\nend","test_suite":"%%\r\nn = 30;\r\nm = 2;\r\ny_correct = '1.00e-02';\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = 30;\r\nm = 8;\r\ny_correct = '7.22e-13';\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = 1;\r\nm = 20;\r\ny_correct = '4.77e-08';\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = 5:5:30;\r\nm = 2;\r\ny_correct = '5.67e-02  2.94e-02  1.98e-02  1.49e-02  1.20e-02  1.00e-02';\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = 10;\r\nm = 2:2:10;\r\ny_correct = ['2.94e-02'; '6.62e-05'; '2.54e-07'; '1.24e-09'; '6.92e-12'];\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = [25 30];\r\nm = [2 8];\r\ny_correct = ['1.20e-02  1.00e-02'; '2.53e-12  7.22e-13'];\r\nassert(isequal(PiByZeta(n,m),y_correct))\r\n\r\n%%\r\nn = 1:5;\r\nm = 2:2:10;\r\ny_correct = ['2.20e-01  1.28e-01  9.04e-02  6.97e-02  5.67e-02'; ...\r\n    '1.96e-02  4.61e-03  1.73e-03  8.26e-04  4.56e-04'; ...\r\n    '2.86e-03  2.82e-04  5.67e-05  1.67e-05  6.25e-06'; ...\r\n    '5.09e-04  2.13e-05  2.33e-06  4.27e-07  1.09e-07'; ...\r\n    '9.94e-05  1.80e-06  1.08e-07  1.24e-08  2.14e-09'];\r\nassert(isequal(PiByZeta(n,m),y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-14T14:03:44.000Z","updated_at":"2026-01-09T15:30:33.000Z","published_at":"2020-06-14T16:19:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problems\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2908\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2909\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e ask us to estimate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by summing a given number of terms in an infinite sum, the Leibniz formula.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHere you are asked to estimate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by summing a given number of terms in other infinite sums corresponding to the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Riemann_zeta_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRiemann zeta function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e evaluated at positive even integers. For example, in solving the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Basel_problem\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eBasel problem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, Euler summed the reciprocals of the squares of positive integers and showed that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"zeta(2) = pi^2/6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(2) = \\\\pi^2/6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with the numbers of terms to sum and a vector \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eabs(E-pi)/pi\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Report the relative error in scientific notation using num2str(...,'%10.2e').\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, PiByZeta(30,2) should return\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ '1.00e-02']]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhile PiByZeta([25 30],[2 8]) should return\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ '1.20e-02  1.00e-02'\\n '2.53e-12  7.22e-13']]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46043,"title":"Evaluate the generalized hypergeometric function","description":"The \u003chttps://en.wikipedia.org/wiki/Generalized_hypergeometric_function generalized hypergeometric function\u003e is defined as \r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/1622e60ecca4a7a8287805cbc798387110f49c68\u003e\u003e\r\n\r\n \r\n \r\nThe numbers _p_ and _q_ are the numbers of values _a_ and _b_ in the numerator and denominator (respectively), and the Pochhammer symbol (a)_n is defined by\r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/c560a95c630b385d8bdf14da55e36d1286d8c68f\u003e\u003e\r\n\r\n`\r\n\r\nMany other functions can be expressed in terms of the generalized hypergeometric function. For example, \r\n\r\n\r\n  exp(x)       = pFq([],[],x)\r\n  cos(x)       = pFq([],1/2,-x^2/4)\r\n  besselj(0,x) = pFq([],1,-x^2/4)\r\n  \r\nThe generalized hypergeometric function can be computed with |hypergeom| from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\r\n\r\nWrite a function to evaluate the generalized hypergeometric function. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 395.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 197.95px; transform-origin: 407px 197.95px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egeneralized hypergeometric function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(See the definition at the Wikipedia link)\" style=\"width: 301px; height: 45px;\" width=\"301\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.4167px; text-align: left; transform-origin: 384px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.8px; transform-origin: 82.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the numbers of values\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.1px 7.8px; transform-origin: 180.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(a)_n\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\" style=\"width: 184px; height: 20px;\" width=\"184\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327.917px 7.8px; transform-origin: 327.917px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"exp(x) = pFq([],[],x)\" style=\"width: 104.5px; height: 21px;\" width=\"104.5\" height=\"21\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUEAAABOCAYAAAC+L6RQAAAK00lEQVR4nO2du3HDPBaFTw/qQA2oAQUbuwLHf6Jg0x13oBZcg0pwD2rBFWzgFrwBdZZXNAACJEAKwPlmOOOxHoRI8OC+AABCCCGEEEIIIYQQQgghhBBCiGo4AbgBOO7dEFGEC4Dr3o0Q4lU5AfgB8L53Q0QxDgDuAD73bogQrwYFUFZC+xww3GsJoRAPjtBD0RInAGeEQxoc9HTPRffQPbo//hb18o5B2H7NcYVfDN8f77ls0johXpQbhgfntHdDxCreAHxjvI9vGAUxZO1dH+85F22dEC+KLIF2uOPvQPaG4f7+wG/lHzCIZ+g9QjQJ44BfezdkJUcomw0AH57/0y0OCdz58Z5b7kYJ8crcULcbdMTg5unhDfOLwdKbo/b+IEQSHPlrzAweMMaxeEgE3ZwwJkfmOCJeMIWonm8MHb7GWSEfGB5uPuASQT8fSIv10bJWeEE0DZMhNVqBUySCflgQ/ZbwGVmDogtoBbYQ+5EI+rlhWdafsUFZg6JJGAu8792QTEgE3XxguaXPshpZg6JJWhvlexDBMwZRc92z98drx8n/XNfjiPjYIL2FFFdaiJfngLiasZpoWQQ/MdRwUpB+MRZDc6oj/0+r7x1uC+6AtHpQZt9bvK6iYy5or2O3LIKEIYxfjAXRXxiE6v3x9xueQx1fk+MHafFBm3lvZcBcxQF555Vqjuo+0HJoaYpcDyIIjHOAbxiEcPp7D4//TcXPHqliRgu0ldDJYmhG5xSuK7Rm3dZYV7jG2kAfvYggY7k/2C5hodk4KCOA5Io26tRiYXB7yZEjOM3awNYyfr2IIEMZW1pmdvGFbrmjrOu0tI6pVg4YRM1O97rjr+hd8Rz0znGNOKq3NvD0IoI2RrdVxtZ6Dy3UlCZzRfnVRbiKSU8xQhvknnNNKZg5OiDjO60NOr2IIC35rQcy9hvfCjXNwqkzW6j/B9op3I3BWoJzrikFc212jvezxRG9BxE8YegrFKQtQxqMRbZ8fZ2wPmkL+ID2koFip4pZ2eOMPAMEYzu/Gb7r1WhdBFkPeMFzXHCr5BYH7a7ignuIElP7PWBd4ZjYTo5QATtyixZ36yL4idH9tXFBPp+l9xC24Zts9YInjAHwC+IU/fh4r80WzjXo8HgfP/P++J65h4qjzZZlFHuccw9sh/Itb8T7lJOvxzlbG2hs4L4VEWTxM6snpgMX6wW/MDw3pV1jK7yrQyknjFNgrhjX+WKg0/VA8DN3jBXh/FyoAtxupXd+HJwGM2fWso0pWMGdXqgDRgH3QXFo3SW28UCXIDFRlLtCn5nmloLbJzyHFlrZMJ4DFmN/077wOfN6CbJUKdhNrW2jbaxmOpKdH59xbYPI7/Nlir7hdn0ooCFSLIYjRleW7bE3xrZzTuRiYmS+Niytv7PHFgkDW/YyFSTGfkpYNL5z1goXDfAdNVcbsD9f4BY4GhW+10vAZ3hx/+HChr49XXmCH8dnQnEja1VYhab56hIy7irlg/HAJWJk23M256IV+IWwm7fUXZuWnCw9SpeOWNeNgxeXO7KDSG5rxp5XK4KIJdA6XTxAUxx8KsoYgH3dZoF8am/LHqywWVFwdfoYt3SJ4tvYwQVjzCIWbv6dSi5LsLT1YK3+n8m57ZSo3KO77Q+tlceIbVgdU+YIn/KQ8aQx8Tt2cPv91gVNMZvXiKA9L+ORKfC3tIqN5bhG1FIZ8lwiuGba35Yhh9LkGHBrg313Uf+01lqKCLpcZBd2xy3rRlmrIyVovFYEbaA6NcNZowjarP3cA27XhHNZyCmW8xnxrq3tC2usTBvueNWQwxasvQY11tvZe5+MHYVTRDD2gvnigsDQ+a1FSCUPtWOtCMa48T6WZKX3hEmmMwYxDCU17GDoGyBunv9Pv4cDTWzcdlUHNuSwBGtOWJC116DGgSCbCKb8eGs1xMwt9cX/XPuzfsPfGRnXWyqC1uVLDfAvNbf3yA7TurL3NJRUsoPDUqE/P75nLxEU/bKqD9nM3NwDzn1UgWe3MuT22MZRLI/4+0BbC2KuLb6Y1Rxc1jtUuhNiyWeAfbLDvj177xh+//T/9tov+Y0W3nOJoNiK1X3I1ob5BI01YnQhY1eMYBzNZlXPgc/YLGSovalZWk7wPptzpFg8tECXuApbZ4fZVtc1pCU8/R02LLG2TKVHEfwPgP8uPP7Zob1L+TeW/87Y418L2rW6D1lBc7mJjCdNXVCKp8uyAPxrfbHI2hWTi9luke5zLBRwO5dxap3eEH74eY1qiBmxrb5i9OnAZUuHlsRKfedoQQQ/MAzknwjf+1IiyFknqeVcpWhWBIHnUhZaSV8I19/Y2RYu95RiNX0Yzp7/A+OPCd1wPrQh0eJagxcMYmDPZR96zn+ccwFvqCcpwmsYEkH7mo3J5ljAoBURnA4OeyzuYMMUOaz0VsnSh7gJiisW9QG/dcC5wxRCzh++wV/2YmNkn3iecxyb9PhGWLhs+10C/T3zuoWzY15hJI4hRQRpldvBb22tXCsiOM2Y7+EJ2ETe0lh4D2TtQ7auLGW1EPu5C8IP0sF8ry1rSDkfXT6fOIc2g2Z7Q69Pz1VipkQpYkVwbn7r2vPXLoLA2I8oRlv3Abu4Bxc2EX955T5UFC7nUxLOL65p9Q+Wx4REsKRF0ZIIkjfsv9YhV2oSf6mhDxVhi/0/rqjPBaEb54ph+rLDOUkVwaUF+1vBxFpqmOCIZ2s7Zo1NH5y/XzMXlDMmGNeuJW6fFZa+lHBTOPrX4gZbQnWCpcVmjQi+2txdCmCKBeaaAMDjJ/G7gFEAa+yHpLQH0uqivNGckb+TMOFTa8cLzRhZWww9R6oIZl0ZODNvWD7HnBUWtg43Nct7QF2hGBf2/koEC3JC3s4SyojXwhXj3GFaNFsI+5K13fiQ1JKB98EaVFetra2+2GrF5b1hTL20CHKgUeJI/MHOVikdb5vuM5OS8bflWDXDRXpDr7+q1VsCFnqvXvR0hlb6j+gYPiSlXfWSxGw/amcr1W71zsHSsiPKiqBvZpoQVcGsdc0xHW4YNveeHkSQlRuMfZYUwVevLhAiil7qvHp5YO94tupLiqBd+0CIarHi0PLeznSH9y68LskH/paWlRRBliPV7EUI0c2Oc8wQ11724oOD2dTKLSmC/G5lhkX1tLgBu4V1mq1aLHYL2yklRbCHwVN0QutuDVdWatXdD+1GWEoEbSF2D3WXonHsrnOtUdOivEuYW2mplAj2EGMVncG1DVtybbj4cEu/ycJymBv8y7KxSPyO50L6tTDG2nK5kegM1gvWXDRtaV0AgeUbia21Cm0yrdUQg+gQusQtLInUgwACcRuJ0RL8xvPSYmsI7aUjRNXwgal5ChS3ZJh70FvNhE8pERNsvdxIdIxrR7ya4Io9c3VrnxHvaYXcIkhXuKZtL4SIxm5wVFsHpwCy1Md3+Ba9bZXcIsiscK0DpRCzMEFSk7toBTDmaLUe0kVuEextEBEdEton5VWxOyfGHK3WDLrgXis5kkRMiMgKFM1Da1CBb2GRFSi6gRlWBb8FoRVYU5hEiFUwAK5OL7hAQy/7swjxfxhUl/vTN1xgo+b6USEWwbmpmhnQL5ya10tdpRB/4HQ6ucX9QTdYg6DoHs4kaX0urniG6zAqDigEhrKZH/RVY9czut9COOCDoURJ23ChVgmgEA4+8XdHM9EOEkAhIjhBItgqRxS8t/8DQB9SykTYg8YAAAAASUVORK5CYII=\" alt=\"cos(x0 = pFq([],1/2,-x^2/4)\" style=\"width: 160.5px; height: 39px;\" width=\"160.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"besselj(0,x) = pFq([],1,-x^2/4)\" style=\"width: 149px; height: 39px;\" width=\"149\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.2333px; text-align: left; transform-origin: 384px 21.2333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 379.9px 7.8px; transform-origin: 379.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.65px 7.8px; transform-origin: 34.65px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 34.65px 8.25px; transform-origin: 34.65px 8.25px; \"\u003ehypergeom\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 252.3px 7.8px; transform-origin: 252.3px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.05px 7.8px; transform-origin: 213.05px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = pFq(a,b,z)\r\n  y = f(a,b,z);\r\nend","test_suite":"%%  exp(x)\r\na = [];\r\nb = [];\r\nz = 1;\r\npFq_correct = exp(z);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  cos(x)\r\na = [];\r\nb = 1/2;\r\nx = pi/4;\r\nz = -x^2/4;\r\npFq_correct = 1/sqrt(2);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  J_0(x)\r\na = [];\r\nb = 1;\r\nx = 1;\r\nz = -x^2/4;\r\npFq_correct = besselj(0,x);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  1/(1-x)^a\r\na = 2;\r\nb = [];\r\nz = 1/2;\r\npFq_correct = 4;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  Example from \"help hypergeom\"--current version of help gives hypergeom(1,2,3) = exp(1)-1\r\na = 1;\r\nb = 2;\r\nz = 3;\r\npFq_correct = (exp(3)-1)/3;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  Hypergeometric function F(a,b; c; x)\r\na = [1 2];\r\nb = 4;\r\nz = 0.2;\r\npFq_correct = 1.113869211474147;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%%  \r\na = [1 1];\r\nb = 2;\r\nz = rand;\r\npFq_correct = -log(1-z)/z;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-11T20:25:32.000Z","updated_at":"2026-01-09T17:23:16.000Z","published_at":"2020-07-11T22:27:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egeneralized hypergeometric function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is defined as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(See the definition at the Wikipedia link)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_pF_q(a_1,\\\\ldots,a_p; c_1,\\\\ldots,c_q; x) = \\\\sum_{n=0}^\\\\infty \\\\frac{(a_1)_n\\\\cdots (a_p)_n}{(c_1)_n\\\\cdots (c_q)_n}\\\\frac{x^n}{n!}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are the numbers of values\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n = a(a+1)\\\\cdots (a+n-1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"exp(x) = pFq([],[],x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee^x = {}_0F_0(-,-,x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"cos(x0 = pFq([],1/2,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\cos(x) = {}_0F_1\\\\left(-,\\\\frac{1}{2},-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"besselj(0,x) = pFq([],1,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eJ_0(x) = {}_0F_1\\\\left(-,1,-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ehypergeom\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47874,"title":"Compute an integral of an exponential function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.3583px 7.91667px; transform-origin: 72.3583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem builds on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 47673\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 225.583px 7.91667px; transform-origin: 225.583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 131px; height: 44px;\" width=\"131\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.8083px 7.91667px; transform-origin: 53.8083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYou may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = IntegralExpFn(a,b,p)\r\n  y = f(a,b,p);\r\nend","test_suite":"%%\r\na = log(2); b = 1; p = 1;\r\ny_correct = 1/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 1;\r\ny_correct = 1;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 2;\r\ny_correct = sqrt(pi)/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 3; b = 1; p = 3;\r\ny_correct = 0.8929795115691813;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 1; b = 1/2; p = 1.5;\r\ny_correct = 0.8278055502117507;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = randi(10); p = 1/2;\r\ny_correct = 2/b^2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 0; b = 4; p = 7/2;\r\ny_correct = 0;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\nfiletext = fileread('IntegralExpFn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-10T04:36:31.000Z","updated_at":"2026-01-09T17:40:58.000Z","published_at":"2020-12-10T05:15:17.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem builds on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 47673\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\int_0^a \\\\exp(-b x^p) dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49743,"title":"Determine aquifer properties: unsteady pump test in a confined aquifer","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 714.15px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 357.075px; transform-origin: 407px 357.075px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.358px 7.79167px; transform-origin: 382.358px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 106.633px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.3167px; text-align: left; transform-origin: 384px 53.3167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.567px 7.79167px; transform-origin: 297.567px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.675px 7.79167px; transform-origin: 67.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABYElEQVRYhe2WYRHDIAyFn4c5wMAMoKAK6qAOcFAL1TAJ84AFNNTC9oPkyBhL27Xwi++utztYeSXJCwCdTidhAUwAFvptJjoDWAG86BlbiTNOiJvW4k8S9q2Fb0i7dq3FByFuW4vPJLwiRqEpnsQf2fiAmIZq1jNIIZ/E2FOMvxCjczmjELjTE5AKj6NSxQULLR4QdxyQik66YKkhHsTiHp/dTbrg8rzfxeIe37ub8ZmSHEMf5fCHRSexeCjMe2XOIVrT0kd4fLtF5SHEh2xOyzenQ6aCXbPbFTLkOVrX4zrJDyCPGI3Ng8lCLybZ9RiDVCdr4R12zmZxbh2hnG8OOV8yuC+UouWw05Za85D5HunhYnLKe9pccfHSESpTkvf80+KG/ujwuzhGlP17WvwM7AJN/JDfj8B+LjWe3dV+Bs3nv1rxZWgdrsrpl8N3fYvoHo94AWl2DZOuqRrqTqezyRvS05ZMfyOVtAAAAABJRU5ErkJggg==\" alt=\"h0\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.467px 7.79167px; transform-origin: 124.467px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. A well turned on and pumped at a rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q0\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.9083px 7.79167px; transform-origin: 38.9083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h(r)\" style=\"width: 29px; height: 18.5px;\" width=\"29\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.79167px; transform-origin: 95.2917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = h0 - h\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3417px 7.79167px; transform-origin: 79.3417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a function of distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.725px 7.79167px; transform-origin: 30.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7px 7.79167px; transform-origin: 7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\" style=\"width: 117px; height: 44px;\" width=\"117\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.9833px 7.79167px; transform-origin: 65.9833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the transmissivity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.15px 7.79167px; transform-origin: 67.15px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the storativity, and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.025px; text-align: left; transform-origin: 384px 18.025px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u = Sr^2/(4Tt)\" style=\"width: 49.5px; height: 36px;\" width=\"49.5\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.917px 7.79167px; transform-origin: 349.917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 347.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 173.733px; text-align: left; transform-origin: 384px 173.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 497px;height: 342px\" 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data-image-state=\"image-loaded\" width=\"497\" height=\"342\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,S] = confinedPumpTest(t,s,Q0,r)\r\n  % t = time, s = drawdown, Q0 = pumping rate, r = distance from well\r\n  T = f1(t,s,Q0,r);\r\n  S = f2(t,s,Q0,r);\r\nend","test_suite":"%%\r\nQ0 = 0.15;                                %  Pumping rate (m3/s)\r\nr  = 100;                                 %  Distance from well (m)\r\nt  = [100 1000 10000 1e5 2e5];            %  Time (s)\r\ns  = [0.0402 1.236 3.664 6.276 7.05];     %  Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2;                     %  Transmissivity (m2/s)\r\nS_correct = 9.864e-4;                     %  Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.30;                                %  Pumping rate (m3/s)\r\nr  = 100;                                 %  Distance from well (m)\r\nt  = [100 1000 10000 1e5 2e5];            %  Time (s)\r\ns  = [0.0804 2.472 7.328 12.552 14.1];    %  Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2;                     %  Transmissivity (m2/s)\r\nS_correct = 9.864e-4;                     %  Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1;                                      %  Pumping rate (m3/s)\r\nr  = 40;                                       %  Distance from well (m)\r\nt  = [300 2000 8000 12000 24000 40000];        %  Time (s)\r\ns  = [0.494 1.749 2.830 3.153 3.709 4.120];    %  Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3;                          %  Transmissivity (m2/s)\r\nS_correct = 3.4e-3;                            %  Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1;                                      %  Pumping rate (m3/s)\r\nr  = 65;                                       %  Distance from well (m)\r\nt  = [300 2000 8000 12000 24000 40000];        %  Time (s)\r\ns  = [0.125 1.050 2.067 2.383 2.931 3.339];    %  Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3;                          %  Transmissivity (m2/s)\r\nS_correct = 3.4e-3;                            %  Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 2e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.05;                                     %  Pumping rate (m3/s)\r\nr  = 5+10*rand;                                %  Distance from well (m)\r\nt  = [4e5 9e5 14e5 19e5 24e5];                 %  Time (s)\r\ns  = [0.859 0.918 0.951 0.973 0.991];          %  Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nlogfit = polyfit(log(t),s,1);                  \r\nTapprox = Q0/(4*pi*logfit(1));                       \r\nSapprox = 2.25*Tapprox*exp(-logfit(2)/logfit(1))/r^2;      \r\nassert(abs(T-Tapprox)/Tapprox \u003c 1e-3 \u0026\u0026 abs(S-Sapprox)/Sapprox \u003c 2e-3)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-04T00:26:53.000Z","updated_at":"2026-01-09T18:01:40.000Z","published_at":"2021-01-04T05:23:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. A well turned on and pumped at a rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h(r)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh(r)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = h0 - h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = h_0-h\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a function of distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"t\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = {Q_0 \\\\over 4 \\\\pi T} \\\\int_u^\\\\infty {e^{-x} \\\\over x} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the transmissivity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the storativity, and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = Sr^2/(4Tt)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = {S r^2 \\\\over 4 T t}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"342\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"497\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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an integral of a product of sinusoids","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150.8px 7.91667px; transform-origin: 150.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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alt=\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\" style=\"width: 139px; height: 44px;\" width=\"139\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.5917px 7.91667px; transform-origin: 96.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are integers. You may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1917px 7.91667px; transform-origin: 99.1917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e but other functions are allowed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = intSinmCosn(m,n)\r\n  y = f(m,n);\r\nend","test_suite":"%%\r\nm = 1;\r\nn = 0;\r\ny_correct = 2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 2;\r\ny_correct = pi/2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 2;\r\ny_correct = 2/3;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 2;\r\ny_correct = pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 2;\r\ny_correct = 4/15;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 4;\r\ny_correct = 3*pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 4;\r\ny_correct = 2/5;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 4;\r\ny_correct = pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 4;\r\ny_correct = 4/35;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 4;\r\ny_correct = 3*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 4;\r\ny_correct = 16/315;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 4;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 4;\r\ny_correct = 32/1155;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 6;\r\ny_correct = 5*pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 6;\r\ny_correct = 2/7;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 6;\r\ny_correct = 5*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 6;\r\ny_correct = 4/63;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 6;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 6;\r\ny_correct = 16/693;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 6;\r\ny_correct = 5*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 6;\r\ny_correct = 32/3003;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 8;\r\ny_correct = 35*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 8;\r\ny_correct = 2/9;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 8;\r\ny_correct = 7*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 8;\r\ny_correct = 4/99;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 8;\r\ny_correct = 7*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 8;\r\ny_correct = 16/1287;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 8;\r\ny_correct = 5*pi/2048;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 8;\r\ny_correct = 32/6435;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2*randi(9);\r\nn = m+1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 22;\r\ny_correct = 2/23;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 28;\r\ny_correct = 2/29;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('intSinmCosn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-23T00:29:43.000Z","updated_at":"2026-02-22T14:27:23.000Z","published_at":"2021-03-23T00:33:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^\\\\pi \\\\sin^mx\\\\cos^nxdx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are integers. You may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e but other functions are allowed.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51975,"title":"Compute a sum of Ramanujan","description":"Srinivasa Ramanujan defined the following function:\r\n\r\nWrite a function to compute  for various values of . See also Cody Problems 45960 and 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.675px 7.79167px; transform-origin: 160.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSrinivasa Ramanujan defined the following function:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\" style=\"width: 162.5px; height: 45px;\" width=\"162.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.9917px 7.79167px; transform-origin: 86.9917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(a)\" style=\"width: 32.5px; height: 18.5px;\" width=\"32.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.5083px 7.79167px; transform-origin: 66.5083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for various values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.79167px; transform-origin: 82.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. See also Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45960\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e45960\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Ramanujanphi(a)\r\n  y = f(a);\r\nend","test_suite":"%%\r\na = 2;\r\ny_correct = 2*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 3;\r\ny_correct = log(3);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 4;\r\ny_correct = (3/2)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 5;\r\nphi = (1+sqrt(5))/2;\r\ny_correct = (sqrt(5)/5)*log(phi)+(1/2)*log(5);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 6;\r\ny_correct = (1/2)*log(3)+(2/3)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 7;\r\ny_correct = (2/7)*(log(14)+2*cos(pi/7)*log(cos(pi/14))+2*log(cos(3*pi/14))*sin(pi/14)-2*log(sin(pi/7))*sin(3*pi/14));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 8;\r\ny_correct = log(2)+(sqrt(2)/8)*log((2+sqrt(2))/(2-sqrt(2)));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 12;\r\ny_correct = (1/2)*log(2)+(1/4)*log(3)+(sqrt(3)/6)*log((sqrt(3)+1)/(sqrt(3)-1));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 18;\r\nb = pi/9;\r\ny_correct = (1/9)*(log(2)+2*log(6)+log(sqrt(3)/2)+2*cos(b)*log(cos(b/2))+2*cos(2*b)*log(cos(b))-2*cos(b)*log(sin(b/2))-2*cos(2*b)*log(sin(b))+2*log(cos(2*b))*sin(b/2)-2*log(sin(2*b))*sin(b/2));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = [11 13 17];\r\nsum_correct = 3.003409919427940;\r\nassert(abs(sum(Ramanujanphi(a))-sum_correct)\u003c1e-14)\r\n\r\n%%\r\na = [36 54 72 100];\r\ny_correct = [1.0000515628258977 1.0000152722224909 1.0000064421348023 1.000002404321212];\r\nk = randi(4);\r\nassert(abs(Ramanujanphi(a(k))-y_correct(k))\u003c1e-14)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-06-05T14:29:40.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-05T14:22:18.000Z","updated_at":"2026-01-20T21:12:40.000Z","published_at":"2021-06-05T14:23:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSrinivasa Ramanujan defined the following function:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a) = 1+2\\\\sum_{k=1}^\\\\infty\\\\frac{1}{(a k)^3 – a k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for various values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. See also Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45960\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45960\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52120,"title":"Compute the fractional derivative","description":"Cody Problem 1370 asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the th derivative as . Then a familiar example from calculus would be . \r\nFractional calculus involves derivatives in which the order  is not an integer. With  and , then \r\n\r\nWrite a function that computes the fractional derivative of order  of an expression of the form\r\n\r\nThe first input to the function will be a 2x matrix in which the first row is the coefficients  and the second row is the exponents . The output should be in a similar form. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 256.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 128.275px; transform-origin: 407px 128.275px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/1370\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 1370\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 7.79167px; transform-origin: 294.442px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.0083px 7.79167px; transform-origin: 49.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth derivative as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^q x^a\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Then a familiar example from calculus would be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^2 x^3 = 6 x\" style=\"width: 65.5px; height: 19px;\" width=\"65.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.725px 7.79167px; transform-origin: 179.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFractional calculus involves derivatives in which the order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.1667px 7.79167px; transform-origin: 71.1667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is not an integer. With \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"q = 1/2\" style=\"width: 51.5px; height: 18.5px;\" width=\"51.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 2\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.4417px 7.79167px; transform-origin: 19.4417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, then \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.4583px; text-align: left; transform-origin: 384px 18.4583px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\" style=\"width: 117px; height: 37px;\" width=\"117\" height=\"37\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 196.292px 7.79167px; transform-origin: 196.292px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that computes the fractional derivative of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.675px 7.79167px; transform-origin: 88.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of an expression of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\" style=\"width: 204px; height: 26px;\" width=\"204\" height=\"26\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.8167px; text-align: left; transform-origin: 384px 21.8167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first input to the function will be a 2x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.775px 7.79167px; transform-origin: 143.775px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e matrix in which the first row is the coefficients \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAoCAYAAAALz1FrAAAA9ElEQVRIie2VbQ2EMAyGHw9zMAMYOAUoOAc4wAEW0IAEPGABDWdh92NtRgjf2y65ZE/CDxi06du3BQqFQqEQjQVqoJXLpgz+AkZgBt5yPwAOaFIk6CTYsHquSWbAxCToJdC4cTbL2RSToJEgH7a1t3jZHldhJLjDy5UFrcIB1c1vDd559dmLI0Gqu3K8CYY4RKt40lSt5FSBmCSXmTieAYO3d5TrWkI16wZW+J6tJ90S5sqdJVCWze8kcY+vbk/vmu3tcEhFWIZXmqkrKMk+20N7mXQ7L7H8wJE6hNnWEARn6b8mi2TqxoGMjVcn3l2ohcI/8AWjzEUn2YFV5AAAAABJRU5ErkJggg==\" alt=\"ci\" style=\"width: 12.5px; height: 20px;\" width=\"12.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83.625px 7.79167px; transform-origin: 83.625px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the second row is the exponents \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAAoCAYAAAAPOoFWAAABF0lEQVRYhe2UUQ3DIBCGPw84mAEMoGAK5qAO6mAW0FAJeJiFaqiF7aF3KWNlkJQt2cKX8FD+9n7uehx0Op1O52ewwACMgJM9J8+2pUkAZjFzgAduwF3WqYXRRYLNgEm0e6R91Cg2ux41OkfB3I5+KujVGGCRQFPmHc16YT/raka2U+e6LIgeMrqlsks1q1wgFx1mzLzjRffvjGoCBcqZa2ZvSxyX8LyjD2yZN/1fqdmF9RpMPJfZ81oFS0WX2sgssJ7cSDC9b5qZlxV3rKPcPE/oyeM1sZVsSfZTrqINNWawlnCUD9K5Vxq+OjObDecchobzsoROlsPzsga9zHvXpjmzmBkqu/EI+r9ufKFBtIsPTZZO5894AM2JYzwwHTn5AAAAAElFTkSuQmCC\" alt=\"ai\" style=\"width: 13.5px; height: 20px;\" width=\"13.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The output should be in a similar form. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = fractDeriv(x,q)\r\n  % x = 2xn matrix with coefficients in the first row and exponents in the second row\r\n  % q = order of the derivative\r\n  % y = output matrix with coefficients in the first row and exponents in the second row\r\n  \r\n  y = q*x^(q-1);\r\nend","test_suite":"%% Example from problem description\r\nx = [1; 2];\r\nq = 1/2;\r\ny_correct = [8/(3*sqrt(pi)); 3/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Constant\r\nc = rand;\r\nx = [c; 0];\r\nq = 1/2;\r\ny_correct = [c/sqrt(pi); -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Polynomial #1\r\nx = [3 -7 4; 2 1 0];\r\nq = 1/2;\r\ny_correct = [[8 -14 4]/sqrt(pi); 3/2 1/2 -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Polynomial #2\r\nx = [1:4; 3:-1:0];\r\nq = 1/3;\r\ny_correct = [1.495438426033838 2.658557201837934 3.323196502297416 2.953952446486593; 8/3 5/3 2/3 -1/3];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Quadratic term\r\nx = [7; 2];\r\nq = 3/2;\r\ny_correct = [28/sqrt(pi); 1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives amounting to a first derivative\r\nq = rand;\r\nx = [6; 5];\r\nyy_correct = [30; 4];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),1-q)-yy_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives undoing each other\r\nq = rand;\r\nx = [5; 2];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),-q)-x)\u003c1e-10))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-27T04:55:37.000Z","updated_at":"2026-01-09T18:33:26.000Z","published_at":"2021-06-27T05:00:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/1370\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 1370\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth derivative as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^q x^a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^q x^a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Then a familiar example from calculus would be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^2 x^3 = 6 x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^2 x^3 = 6 x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFractional calculus involves derivatives in which the order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is not an integer. With \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = 1/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = 1/2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^{1/2} x^2 = \\\\frac{8}{3\\\\sqrt{\\\\pi}} x^{3/2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that computes the fractional derivative of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of an expression of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(x) = c­_1 x^{a_1} + c_2 x^{a_2} + c_3 x^{a_3} + \\\\dots \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first input to the function will be a 2x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e matrix in which the first row is the coefficients \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ci\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the second row is the exponents \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ai\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The output should be in a similar form. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52125,"title":"Compute the sum of reciprocals of quadratics","description":"Write a function to compute the following sum:\r\n\r\nSee also Cody Problem 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following sum:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = sum(1/(n^2+an+b),{n,1,infinity})\" style=\"width: 122.5px; height: 45px;\" width=\"122.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.79167px; transform-origin: 29.175px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumRecipQuad(a,b)\r\n  y = sum(1/(n^2+a*n+b));\r\nend","test_suite":"%%\r\na = 3;\r\nb = 2;\r\ny_correct = 1/2;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 5;\r\nb = 6;\r\ny_correct = 1/3;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 7;\r\nb = 10;\r\ny_correct = 47/180;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 8;\r\nb = 15;\r\ny_correct = 9/40;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 10;\r\nb = 21;\r\ny_correct = 319/1680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 9;\r\nb = 14;\r\ny_correct = 153/700;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 13;\r\nb = 22;\r\ny_correct = 42131/249480;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 14;\r\nb = 33;\r\ny_correct = 32891/221760;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 12;\r\nb = 35;\r\ny_correct = 13/84;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 55;\r\ny_correct = 20417/166320;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 15;\r\nb = 26;\r\ny_correct = 605453/3963960;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 39;\r\ny_correct = 485333/3603600;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 18;\r\nb = 65;\r\ny_correct = 323171/2882880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 20;\r\nb = 91;\r\ny_correct = 30233/308880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 143;\r\ny_correct = 25/312;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 21;\r\nb = 38;\r\ny_correct = 158899519/1319157840;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 95;\r\ny_correct = 19622959/217273056;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 30;\r\nb = 209;\r\ny_correct = 11171129/169303680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 36;\r\nb = 323;\r\ny_correct = 37/684;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 25;\r\nb = 46;\r\ny_correct = 265842403/2498640144;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nc = randi(50);\r\na = 2*c+1;\r\nb = c*(c+1);\r\ny_correct = 1/(c+1);\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nfiletext = fileread('sumRecipQuad.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'assignin'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2021-07-03T14:09:11.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2021-06-27T14:48:37.000Z","updated_at":"2026-01-09T19:12:11.000Z","published_at":"2021-06-27T14:53:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following sum:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum(1/(n^2+an+b),{n,1,infinity})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{n=1}^\\\\infty \\\\frac{1}{n^2+an+b}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54610,"title":"Evaluate the Kelvin functions","description":"The Kelvin functions ber, bei, ker, and kei are related to Bessel functions of order . When the order is not specified, the default is . The functions ker() and kei() appear in the solution for velocity in the boundary layer under water waves. \r\nWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57.375px; transform-origin: 407px 57.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.875px; text-align: left; transform-origin: 384px 31.875px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.075px 7.50833px; transform-origin: 75.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Kelvin functions ber\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.225px 7.50833px; transform-origin: 13.225px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, bei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6083px 7.50833px; transform-origin: 13.6083px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, ker\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.45px 7.50833px; transform-origin: 26.45px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and kei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.075px 7.50833px; transform-origin: 124.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are related to Bessel functions of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.725px 7.50833px; transform-origin: 72.725px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. When the order is not specified, the default is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"nu = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.8917px 7.50833px; transform-origin: 59.8917px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The functions ker(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.50833px; transform-origin: 29.175px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and kei(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 175.825px 7.50833px; transform-origin: 175.825px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.867px 7.50833px; transform-origin: 372.867px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ber,bei,ker,kei] = kelvin(x,nu)\r\n  ber = besselr(nu,x);\r\n  bei = besseli(nu,x);\r\n  ker = kesselr(nu,x);\r\n  kei = kesseli(nu,x);\r\nend","test_suite":"%%\r\nx = 3;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-0.221380249598694,1.937586785266043,-0.067029233303799,-0.051121884045987);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = 1:4;\r\ntol = 1e-15;\r\nber = kelvin(x);\r\nber_correct = [0.984381781213087 0.7517341827138085 -0.221380249598694 -2.563416557258581];\r\nassert(all(abs(ber-ber_correct)\u003ctol))\r\n\r\n%%\r\nx = -0.4;\r\nnu = 1;\r\ntol = 1e-15;\r\n[ber,bei] = kelvin(x,nu);\r\n[ber_correct,bei_correct] = deal(0.1442308644531633,-0.1385741359112079);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol)\r\n\r\n%%\r\nx = 3.4;\r\nnu = 0.5;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-2.245652084214816,0.82701468622679,-0.05553905648843065,0.02619201937598225);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = [-psi(1) exp(1) pi];\r\nnu = 1/3;\r\ntol = 2e-14;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\nber_correct = [0.4900252831480887 -0.7214812981316725  -1.432808723213069];\r\nbei_correct = [0.5553968849316957  1.532550247673432    1.535388618042967];\r\nker_correct = [0.2909569660163571 -0.1038243609935959  -0.07552171571938433];\r\nkei_correct = [-0.983688891425642 -0.03413110664720041 -0.001259453778330345];\r\nassert(all(abs(ber-ber_correct)\u003ctol) \u0026 all(abs(bei-bei_correct)\u003ctol) \u0026 all(abs(ker-ker_correct)\u003ctol) \u0026 all(abs(kei-kei_correct)\u003ctol))\r\n\r\n%%\r\nfiletext = fileread('kelvin.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'import'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-06T01:51:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-06T01:49:11.000Z","updated_at":"2026-01-09T20:00:53.000Z","published_at":"2022-05-06T01:51:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Kelvin functions ber\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, bei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, ker\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and kei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are related to Bessel functions of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. When the order is not specified, the default is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The functions ker(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and kei(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54710,"title":"Compute the period of a pendulum started from a finite initial angle","description":"Cody Problem 49830 asks for the period  of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle  that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \r\nWrite a function that takes the initial angle and returns , where  is the length of the pendulum and  is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49830\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 49830\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for the period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"theta0\" style=\"width: 15px; height: 20px;\" width=\"15\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the initial angle and returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T\\sqrt{g/L}\" style=\"width: 52.5px; height: 20.5px;\" width=\"52.5\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the length of the pendulum and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eg\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2pi\" style=\"width: 19.5px; height: 18px;\" width=\"19.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pendulumPeriod(theta0)\r\n  T = theta0-theta0^3/3!+theta0^5/5!+higher order terms;\r\nend","test_suite":"%%\r\nth = pi/7;\r\nT_correct = 6.363207946270837;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/5;\r\nT_correct = 6.44181661515865;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% Problem 1 on p. 194 of Davis (1962)\r\nth = pi/4;\r\nT_correct = 6.534345229832591;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/3;\r\nT_correct = 6.743001419250384;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/2;\r\nT_correct = 7.416298709205487;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = 36*pi/37;\r\nT_correct = 18.190113206504414;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 72*pi/73;\r\nT_correct = 20.902949604823448;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 0;\r\nT_correct = 2*pi;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('pendulumPeriod.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-06T01:13:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-06-06T00:37:45.000Z","updated_at":"2026-01-09T20:11:24.000Z","published_at":"2022-06-06T01:13:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49830\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 49830\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for the period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"theta0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the initial angle and returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\\sqrt{g/L}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\\\\sqrt{g/L}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"L\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the length of the pendulum and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"g\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"no_progress_badge":{"id":53,"name":"Unknown","symbol":"unknown","description":"Partially completed groups","description_html":null,"image_location":"/images/responsive/supporting/matlabcentral/cody/badges/problem_groups_unknown_2.png","bonus":null,"players_count":0,"active":false,"created_by":null,"updated_by":null,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"created_at":"2018-01-10T23:20:29.000Z","updated_at":"2018-01-10T23:20:29.000Z","community_badge_id":null,"award_multiples":false}}