Plot of the Kelvin function ber(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In applied mathematics, the Kelvin functions berν x ) and beiν x ) are the real and imaginary parts , respectively, of
J ν ( x e 3 π i 4 ) , {\displaystyle J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right),\,} where x is real, and Jν (z )ν th order Bessel function of the first kind. Similarly, the functions kerν (x ) and keiν (x ) are the real and imaginary parts, respectively, of
K ν ( x e π i 4 ) , {\displaystyle K_{\nu }\left(xe^{\frac {\pi i}{4}}\right),\,} where Kν (z )ν th order modified Bessel function of the second kind.
These functions are named after William Thomson, 1st Baron Kelvin .
While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments xe iφ φ < 2π .n x ) and bein x ) for integral n , the Kelvin functions have a branch point at x = 0.
Below, Γ(z ) is the gamma function and ψ (z )digamma function .
ber(x ) for x between 0 and 20. b e r ( x ) / e x / 2 {\displaystyle \mathrm {ber} (x)/e^{x/{\sqrt {2}}}} x between 0 and 50.For integers n , bern x ) has the series expansion
b e r n ( x ) = ( x 2 ) n ∑ k ≥ 0 cos [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k , {\displaystyle \mathrm {ber} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k},} where Γ(z ) is the gamma function . The special case ber0 (x ), commonly denoted as just ber(x ), has the series expansion
b e r ( x ) = 1 + ∑ k ≥ 1 ( − 1 ) k [ ( 2 k ) ! ] 2 ( x 2 ) 4 k {\displaystyle \mathrm {ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}}{[(2k)!]^{2}}}\left({\frac {x}{2}}\right)^{4k}} and asymptotic series
b e r ( x ) ∼ e x 2 2 π x ( f 1 ( x ) cos α + g 1 ( x ) sin α ) − k e i ( x ) π {\displaystyle \mathrm {ber} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}\left(f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha \right)-{\frac {\mathrm {kei} (x)}{\pi }}} where
α = x 2 − π 8 , {\displaystyle \alpha ={\frac {x}{\sqrt {2}}}-{\frac {\pi }{8}},} f 1 ( x ) = 1 + ∑ k ≥ 1 cos ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 {\displaystyle f_{1}(x)=1+\sum _{k\geq 1}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}} g 1 ( x ) = ∑ k ≥ 1 sin ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 . {\displaystyle g_{1}(x)=\sum _{k\geq 1}{\frac {\sin(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}.} Plot of the Kelvin function bei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D bei(x ) for x between 0 and 20. b e i ( x ) / e x / 2 {\displaystyle \mathrm {bei} (x)/e^{x/{\sqrt {2}}}} x between 0 and 50.For integers n , bein x ) has the series expansion
b e i n ( x ) = ( x 2 ) n ∑ k ≥ 0 sin [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k . {\displaystyle \mathrm {bei} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k}.} The special case bei0 (x ), commonly denoted as just bei(x ), has the series expansion
Plot of the Kelvin function ker(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D b e i ( x ) = ∑ k ≥ 0 ( − 1 ) k [ ( 2 k + 1 ) ! ] 2 ( x 2 ) 4 k + 2 {\displaystyle \mathrm {bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}}{[(2k+1)!]^{2}}}\left({\frac {x}{2}}\right)^{4k+2}} and asymptotic series
b e i ( x ) ∼ e x 2 2 π x [ f 1 ( x ) sin α − g 1 ( x ) cos α ] − k e r ( x ) π , {\displaystyle \mathrm {bei} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\sin \alpha -g_{1}(x)\cos \alpha ]-{\frac {\mathrm {ker} (x)}{\pi }},} where α, f 1 ( x ) {\displaystyle f_{1}(x)} g 1 ( x ) {\displaystyle g_{1}(x)} x ).
ker(x ) for x between 0 and 14. k e r ( x ) e x / 2 {\displaystyle \mathrm {ker} (x)e^{x/{\sqrt {2}}}} x between 0 and 50.For integers n , kern x ) has the (complicated) series expansion
k e r n ( x ) = − ln ( x 2 ) b e r n ( x ) + π 4 b e i n ( x ) + 1 2 ( x 2 ) − n ∑ k = 0 n − 1 cos [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 cos [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k . {\displaystyle {\begin{aligned}&\mathrm {ker} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}} Plot of the Kelvin function kei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D The special case ker0 (x ), commonly denoted as just ker(x ), has the series expansion
k e r ( x ) = − ln ( x 2 ) b e r ( x ) + π 4 b e i ( x ) + ∑ k ≥ 0 ( − 1 ) k ψ ( 2 k + 1 ) [ ( 2 k ) ! ] 2 ( x 2 4 ) 2 k {\displaystyle \mathrm {ker} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} (x)+{\frac {\pi }{4}}\mathrm {bei} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+1)}{[(2k)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k}} and the asymptotic series
k e r ( x ) ∼ π 2 x e − x 2 [ f 2 ( x ) cos β + g 2 ( x ) sin β ] , {\displaystyle \mathrm {ker} (x)\sim {\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\cos \beta +g_{2}(x)\sin \beta ],} where
β = x 2 + π 8 , {\displaystyle \beta ={\frac {x}{\sqrt {2}}}+{\frac {\pi }{8}},} f 2 ( x ) = 1 + ∑ k ≥ 1 ( − 1 ) k cos ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 {\displaystyle f_{2}(x)=1+\sum _{k\geq 1}(-1)^{k}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}} g 2 ( x ) = ∑ k ≥ 1 ( − 1 ) k sin ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 . {\displaystyle g_{2}(x)=\sum _{k\geq 1}(-1)^{k}{\frac {\sin(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}.}
kei(x ) for x between 0 and 14. k e i ( x ) e x / 2 {\displaystyle \mathrm {kei} (x)e^{x/{\sqrt {2}}}} x between 0 and 50.For integer n , kein x ) has the series expansion
k e i n ( x ) = − ln ( x 2 ) b e i n ( x ) − π 4 b e r n ( x ) − 1 2 ( x 2 ) − n ∑ k = 0 n − 1 sin [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 sin [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k . {\displaystyle {\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}} The special case kei0 (x ), commonly denoted as just kei(x ), has the series expansion
k e i ( x ) = − ln ( x 2 ) b e i ( x ) − π 4 b e r ( x ) + ∑ k ≥ 0 ( − 1 ) k ψ ( 2 k + 2 ) [ ( 2 k + 1 ) ! ] 2 ( x 2 4 ) 2 k + 1 {\displaystyle \mathrm {kei} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} (x)-{\frac {\pi }{4}}\mathrm {ber} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+2)}{[(2k+1)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k+1}} and the asymptotic series
k e i ( x ) ∼ − π 2 x e − x 2 [ f 2 ( x ) sin β + g 2 ( x ) cos β ] , {\displaystyle \mathrm {kei} (x)\sim -{\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\sin \beta +g_{2}(x)\cos \beta ],} where β , f 2 (x ), and g 2 (x ) are defined as for ker(x ).
Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 9" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-61272-0 LCCN 64-60036 . MR 0167642 . LCCN 65-12253 .Olver, F. W. J.; Maximon, L. C. (2010), "Bessel functions" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. [1] GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [2] 
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![{\displaystyle \mathrm {ber} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0cd93458b54421e6ad2cc1a1d7642427b33221)
![{\displaystyle \mathrm {ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}}{[(2k)!]^{2}}}\left({\frac {x}{2}}\right)^{4k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0382d07ebb15b81d9f6d57b4a4c39d727bd7dd)
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![{\displaystyle \mathrm {bei} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c08343a9ce9fef41ac36a7add2de1ef813cc3ec3)
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![{\displaystyle \mathrm {bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}}{[(2k+1)!]^{2}}}\left({\frac {x}{2}}\right)^{4k+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f69dab8cb68e3d5e5a8dccfdf3f6c798775a0490)
![{\displaystyle \mathrm {bei} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\sin \alpha -g_{1}(x)\cos \alpha ]-{\frac {\mathrm {ker} (x)}{\pi }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64ac7c7ceb46b3e4f8fe6307064ff05eeb980e04)
![{\displaystyle {\begin{aligned}&\mathrm {ker} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c861946c574b893b798807b4d330a08838a82c03)
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![{\displaystyle \mathrm {ker} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} (x)+{\frac {\pi }{4}}\mathrm {bei} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+1)}{[(2k)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e620fcd5ca5f6cba6b4ba993439a8e6878b3505c)
![{\displaystyle \mathrm {ker} (x)\sim {\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\cos \beta +g_{2}(x)\sin \beta ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6156e93597225fc773d61fff4b8c7b9b2ddc16f2)
![{\displaystyle {\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a32a3a0932238e7433d48965287b3ab97f0f245)
![{\displaystyle \mathrm {kei} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} (x)-{\frac {\pi }{4}}\mathrm {ber} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+2)}{[(2k+1)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a84894452aa4ac878be50e50a42b2a91bd3bce86)
![{\displaystyle \mathrm {kei} (x)\sim -{\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\sin \beta +g_{2}(x)\cos \beta ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb3fd63059e08690345a00b76f7425ba1bcc768)