Problem 45982. Evaluate the incomplete elliptic integrals

Created by ChrisR

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<div 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; "><div style="block-size: 165.05px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.525px; transform-origin: 407px 82.525px; vertical-align: baseline; "><div style="block-size: 63.05px; 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 31.525px; text-align: left; transform-origin: 384px 31.525px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 256.742px 7.79167px; transform-origin: 256.742px 7.79167px; unicode-bidi: normal; ">Elliptic integrals can be used to evaluate integrals whose integrands have the form <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ4AAAApCAYAAADESpC2AAAFVklEQVR4nO1by7XiSgysHMiABLyaHYu3JgIyIAMyIAW2b0sI5EAKRPAWpPBmYepY1u2P1N3md7vO8Zk7YNyyulQtqW2go6Ojo6OjFkcAw6uN6Phd2AD4H8D61YZ0/C5cAPwL4L8vP/5p5K+OBhgwqt0fvJ4YnXi/CKfH0dHxNKzRc7uPwArfVfm9Uu0GjP5sgbXzWh81hyuMSfhHGZ0A1e4V97NDW8IP8M3N8XE0xQbAwXBsHdf8NtIB48RfMufskPfjHj6/tCYdsQZwc9hyXMKOAcAVY0TzuGJ01BHA/fHZHTbmXzE6+FtAtdsYzt1i8hePE0ZfnsRnN4ykyl3rhnZLrMbgvP4ZC8zrAZNT7uq7FebETDH/iLwyfBosaidxQdxXA+bEjJFv/TjPQvYaHDESygLa1HQlO2JyRsiQLeZRHBrcowyfghX8BLghTSzp65jieMleCt6fNZU6YBShZpDOismpJF7Ioc9y1jNxgO+eGHw8Qq2XjTpHE+/ZAXzCOP8W0LZcmuC6WMpZUOfoCGlq0JugRO12mKtZCJp4evXwEKEFuBtjvc8zGgmMxVl6qdXk3Ec+/2SULCuygIjlwjnFuyd+uxQ8Yzab65yzdHERqmwvKItStnNCSsn2RAsybzA613otqp1XwWXhEMubZI6nlYOk9I47YCTEIfAdfZxSNM/8ldr4AylnrTEnXawCCjkxBeaDMrfkkuOpoi04YLpH67UO8AcSl6yYkgFTZchzNBkOkc9Tdp4x95fM0aWoXCM2yXGtrZWYAJmhnXXE1L+TN5Pqza1Rbohcdhitl8e1do+/Pc1rjfPj91weLM4tVTvZkgot0bpfGvInieLt3dFmKQAsjLYY/RBSQ227lfBeofkBOSE3TMbKpSC3PJE8qRtLgQ6jc6x9JQtW4l+Ok7Nzh7K0QfqNDXge+rvYBPO8EpwxzaO3+cwc3hrkV1S2VaRDGIFShSzG1xKPDrtj2WqOUX1H+p4suwoaK8xXjtB22SYzLlBHPC0inkavdw5r7AQQL+tzTVCJWuJJhy3ZjpFto9g4pWonq/4csVOomVCZNnnz4qcSTyqb3iaT+Upu6aslnnRYTT5nAXOoGLluKNuPzO38WFGrJEwnvPm2t1It7WIASDvL2lQGJuKUEk/2EZfuX6VUb4dytbLs/FjAtKMEch/YS4qnFhe5CiuU/6UMKYl0PiHBiXtGx573pR3H4soLT5DmQAJ4N+LZgpJz6rGD41p/UywSOhkODSiVKFfBlFQ5dNYe8zxv6d0PmWIwwmvUTif1NfBWl8QJY+DLtMWTL5/xM92KgWMUKbtlm0y2IHJRyGXbA/koechhe8xzlS1GlWJvrgZUhrP4f21V3iJVoCB4bNEFkW6Ws3+Ywg32FYvcKXo8Sne0LeelElYSJ0UINoT5NKseVzZA9/gZEDIIPNtfMVsk0UvVTgdni+KIOxEx8NF1PmCqfSH7eSRliiScO6tCnlGo7HJpyPV89IZ2yrE3pCNe5oyh/qB+Qld/LxP42iReX69E7VaY20wFrX1imEERCyz90K4uCHaZ70PXsy6zDDS371PvA8QYbz2PNxxz/BrTewehc1aY8r3YPidtaKEunm20lC2hozZPvSG+wtBPqXE8D1h4iqqa1WFRcCldEsyDam+e0bu0vSXwPk1TilBKE8MKZbs6T8Eiz+UrMLdpAe87ps+E532IEnjnaml7quF9e8mDE95ToZbCIm92PXCFXb22SD9W9TbYYFx2Wxo64LteJLKABUzr5c1zTVbRb086YsCb5gMfiAPaTfwWvgBuOXZHR0dHR0dHR0dHR4cdfwGlCMZI4WYYjwAAAABJRU5ErkJggg==" alt="R(x,sqrt(P(x)))" style="width: 79px; height: 20.5px;" width="79" height="20.5"><span 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; ">, where <img src="data:image/png;base64,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" alt="R(x,y)" style="width: 46px; height: 18.5px;" width="46" height="18.5"><span 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: 39.675px 7.79167px; transform-origin: 39.675px 7.79167px; unicode-bidi: normal; "> is a rational function in x<span 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; "> and y<span 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; "> and <img src="data:image/png;base64,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" alt="P(x)" style="width: 31.5px; height: 18.5px;" width="31.5" height="18.5"><span 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: 254.792px 7.79167px; transform-origin: 254.792px 7.79167px; unicode-bidi: normal; "> is a polynomial of degree 4 or less. They appear in calculations of arclength (as in<span 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; "> <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse">Cody Problem 45979</a><span 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: 133.808px 7.79167px; transform-origin: 133.808px 7.79167px; unicode-bidi: normal; ">) and analysis of the motion of a pendulum.</div><div 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; "><span 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: 342.542px 7.79167px; transform-origin: 342.542px 7.79167px; unicode-bidi: normal; ">MATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the <a target='_blank' href = "https://en.wikipedia.org/wiki/Elliptic_integral">incomplete elliptic integrals</a><span 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: 110.45px 7.79167px; transform-origin: 110.45px 7.79167px; unicode-bidi: normal; "> of the first, second, and third kinds.</div><div 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; "><span 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: 367.883px 7.79167px; transform-origin: 367.883px 7.79167px; unicode-bidi: normal; ">Write a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter m<span 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; ">, which is related to the modulus k<span 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: 27.6167px 7.79167px; transform-origin: 27.6167px 7.79167px; unicode-bidi: normal; "> through <img src="data:image/png;base64,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" alt="m = k^2" style="width: 44.5px; height: 19px;" width="44.5" height="19"><span 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; ">.</div></div></div>

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Problem 45982. Evaluate the incomplete elliptic integrals

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