Problem 54610. Evaluate the Kelvin functions

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: 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; "><div 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; "><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: 75.075px 7.50833px; transform-origin: 75.075px 7.50833px; unicode-bidi: normal; ">The Kelvin functions ber<img src="data:image/png;base64,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" alt="_nu(x)" style="width: 27px; height: 20px;" width="27" height="20"><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: 13.225px 7.50833px; transform-origin: 13.225px 7.50833px; unicode-bidi: normal; ">, bei<img src="data:image/png;base64,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" alt="_nu(x)" style="width: 27px; height: 20px;" width="27" height="20"><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: 13.6083px 7.50833px; transform-origin: 13.6083px 7.50833px; unicode-bidi: normal; ">, ker<img src="data:image/png;base64,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" alt="_nu(x)" style="width: 27px; height: 20px;" width="27" height="20"><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: 26.45px 7.50833px; transform-origin: 26.45px 7.50833px; unicode-bidi: normal; ">, and kei<img src="data:image/png;base64,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" alt="_nu(x)" style="width: 27px; height: 20px;" width="27" height="20"><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: 124.075px 7.50833px; transform-origin: 124.075px 7.50833px; unicode-bidi: normal; "> are related to Bessel functions of order ν<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: 72.725px 7.50833px; transform-origin: 72.725px 7.50833px; unicode-bidi: normal; ">. When the order is not specified, the default is <img src="data:image/png;base64,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" alt="nu = 0" style="width: 36px; height: 18px;" width="36" height="18"><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: 59.8917px 7.50833px; transform-origin: 59.8917px 7.50833px; unicode-bidi: normal; ">. The functions ker(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: 29.175px 7.50833px; transform-origin: 29.175px 7.50833px; unicode-bidi: normal; ">) and kei(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: 175.825px 7.50833px; transform-origin: 175.825px 7.50833px; unicode-bidi: normal; ">) appear in the solution for velocity in the boundary layer under water waves. </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: 372.867px 7.50833px; transform-origin: 372.867px 7.50833px; unicode-bidi: normal; ">Write 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. </div></div></div>

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ChrisR on 6 May 2022

For what it's worth, this is the solution that caused William to break the 50,000 point barrier.

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Problem 54610. Evaluate the Kelvin functions

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