{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"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,iVBORw0KGgoAAAANSUhEUgAAADYAAAAoCAYAAAC1mQk2AAADWUlEQVRoQ+2YO89NQRSGz/cDRFwqlbgkOgqiQaFASESiQLRCKFRCoZS4hehQKBQSvohGIShIUBAkVAqXjsol4gfwPjIrmUxmZs+cPWeTL2cnb07Onpk16123WbNnRnP0mZmjvEZTYpWe3a75n4W3leti05fp5SrhXo2sSXhsrxTYJBypUaRj7nGNL62R2ZoYpHYJ/LZ+LjuBRQZrSWyDNr4rrBS+tWbl5N3X73XhVpf8VsQWaaP3LlQ6N+1SKjO+WmOPhbXCp5ycVsTIgYPCih5Kly4lJCko2yZNbChvGQ9C/qmwRkhW3RYeO6QNrgqLJ5hboXM+6MVDF/pRx7Ug9tJJXlcaSw3mEY5U3oUpWX2JEYZfhdPCyQYKl4qwKEmGY19iFu8npNH5Uq3cPArAEuFdJISR+yuTQ7bvPs2JVuG+xMxyG7XBswJilGvWbBGWu/m+txm/48Z+6DdV1jsN2pcYZf6cUEoML/10HqIAQO6jsF6YL7wSrgh7BPInmUMa++3mRjuRoYn5TjWj8A6Cs8LFVGhFogFiD4ToefYviVk4oTOeonzXFKD/lhiEUI6HfMqFXeiwzmrc12OcJTeF0hwLFaSp3eo8VnMOTrx4dG4QyQ3/FaWaQoHHam4Ftu8OrYteQPt6zMKJSlZ0T/JY2VFhr5JKRoxjhYeiE+3yWxDD6mxQE0qcV2/cOso9T033wp6cccnbRIqYfWdgQ9/VJC3njW8ly7Ok9QKLI+OFcNTJjuUZveCFlDf0nqKT7XZixAiRM8KCyGK78vttjF1bzmp+qq1CJs9z4Zpw25sbnmfHNEbOpUp/kSFTHjMrhk0mnfwBIbwH5S6a/nkFOQ5i/5uIhaU5NhwPHD5Ct9cZ4n/nx4jZGUHs+zGMgpeEVC5B2veEKYS8UwLh/SjhVQyzOTNusvisd0PorKAxYubqMJlRPNfyWAO7O+LR0Orj/LeQ36nFnQ13jBiJe1jwyy/v8FRX5TOv0r+1/lJV/IUqFYrfnTlxNw9hxDUDZL8Mufl4jo5/f0NyGPaJUPwFLPSYJbI1pXgOorXhRT7NaxSSY8kKifmll+KR/WAyTqIMtSYkZtf1L4VhN5Se1fu0aKmqNx1iwZTYEFZuucfUYy2tOYSsqceGsHLLPf4AnTK3KUKaO2cAAAAASUVORK5CYII=\" 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\"}]}"}],"problem_search":{"errors":[],"problems":[{"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,iVBORw0KGgoAAAANSUhEUgAAADYAAAAoCAYAAAC1mQk2AAADWUlEQVRoQ+2YO89NQRSGz/cDRFwqlbgkOgqiQaFASESiQLRCKFRCoZS4hehQKBQSvohGIShIUBAkVAqXjsol4gfwPjIrmUxmZs+cPWeTL2cnb07Onpk16123WbNnRnP0mZmjvEZTYpWe3a75n4W3leti05fp5SrhXo2sSXhsrxTYJBypUaRj7nGNL62R2ZoYpHYJ/LZ+LjuBRQZrSWyDNr4rrBS+tWbl5N3X73XhVpf8VsQWaaP3LlQ6N+1SKjO+WmOPhbXCp5ycVsTIgYPCih5Kly4lJCko2yZNbChvGQ9C/qmwRkhW3RYeO6QNrgqLJ5hboXM+6MVDF/pRx7Ug9tJJXlcaSw3mEY5U3oUpWX2JEYZfhdPCyQYKl4qwKEmGY19iFu8npNH5Uq3cPArAEuFdJISR+yuTQ7bvPs2JVuG+xMxyG7XBswJilGvWbBGWu/m+txm/48Z+6DdV1jsN2pcYZf6cUEoML/10HqIAQO6jsF6YL7wSrgh7BPInmUMa++3mRjuRoYn5TjWj8A6Cs8LFVGhFogFiD4ToefYviVk4oTOeonzXFKD/lhiEUI6HfMqFXeiwzmrc12OcJTeF0hwLFaSp3eo8VnMOTrx4dG4QyQ3/FaWaQoHHam4Ftu8OrYteQPt6zMKJSlZ0T/JY2VFhr5JKRoxjhYeiE+3yWxDD6mxQE0qcV2/cOso9T033wp6cccnbRIqYfWdgQ9/VJC3njW8ly7Ok9QKLI+OFcNTJjuUZveCFlDf0nqKT7XZixAiRM8KCyGK78vttjF1bzmp+qq1CJs9z4Zpw25sbnmfHNEbOpUp/kSFTHjMrhk0mnfwBIbwH5S6a/nkFOQ5i/5uIhaU5NhwPHD5Ct9cZ4n/nx4jZGUHs+zGMgpeEVC5B2veEKYS8UwLh/SjhVQyzOTNusvisd0PorKAxYubqMJlRPNfyWAO7O+LR0Orj/LeQ36nFnQ13jBiJe1jwyy/v8FRX5TOv0r+1/lJV/IUqFYrfnTlxNw9hxDUDZL8Mufl4jo5/f0NyGPaJUPwFLPSYJbI1pXgOorXhRT7NaxSSY8kKifmll+KR/WAyTqIMtSYkZtf1L4VhN5Se1fu0aKmqNx1iwZTYEFZuucfUYy2tOYSsqceGsHLLPf4AnTK3KUKaO2cAAAAASUVORK5CYII=\" 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\"}]}"}],"term":"tag:\"bessel 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