In applied mathematics, the Kelvin functions ber ν x ) and bei ν x ) are the real and imaginary parts , respectively, of
J
ν
(
x
e
3
π
i
/
4
)
,
{\displaystyle J_{\nu }(xe^{3\pi i/4}),\,}
where x is real, and J ν z ),
is the νth 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 }(xe^{\pi i/4})\,}
K
ν
(
z
)
{\displaystyle K_{\nu }(z)\,}
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 x e i φn x ) and Bein x ) for integral n , the Kelvin functions have a branch point at x = 0.
x )ber(x ) for x between 0 and 10.
b
e
r
(
x
)
/
e
x
/
2
{\displaystyle \mathrm {ber} (x)/e^{x/{\sqrt {2}}}}
x
{\displaystyle x}
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 ber 0 x ), commonly denoted as just ber (x ), has the series expansion
b
e
r
(
x
)
=
1
+
∑
k
≥
1
(
−
1
)
k
(
x
/
2
)
4
k
[
(
2
k
)
!
]
2
{\displaystyle \mathrm {ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}(x/2)^{4k}}{[(2k)!]^{2}}}}
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}}}[f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha ]-{\frac {\mathrm {kei} (x)}{\pi }}}
where
α
=
x
/
2
−
π
/
8
{\displaystyle \alpha =x/{\sqrt {2}}-\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}}
x )bei(x) for
x
{\displaystyle x}
b
e
i
(
x
)
/
e
x
/
2
{\displaystyle \mathrm {bei} (x)/e^{x/{\sqrt {2}}}}
x
{\displaystyle x}
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}}
where Γ(z ) is the Gamma function . The special case bei0 (x), commonly denoted as just bei(x), has the series expansion
b
e
i
(
x
)
=
∑
k
≥
0
(
−
1
)
k
(
x
/
2
)
4
k
+
2
[
(
2
k
+
1
)
!
]
2
{\displaystyle \mathrm {bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}(x/2)^{4k+2}}{[(2k+1)!]^{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
{\displaystyle \alpha }
f
1
(
x
)
{\displaystyle f_{1}(x)}
g
1
(
x
)
{\displaystyle g_{1}(x)}
(
x
)
{\displaystyle (x)}
x )For integers n , kern x ) has the (complicated) series expansion
k
e
r
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
−
ln
(
x
2
)
b
e
r
n
(
x
)
+
π
4
b
e
i
n
(
x
)
+
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)&={\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}-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&{}\quad +{\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}}}
ker(x) for
x
{\displaystyle x}
k
e
r
(
x
)
e
x
/
2
{\displaystyle \mathrm {ker} (x)e^{x/{\sqrt {2}}}}
x between 0 and 100.where
ψ
(
z
)
{\displaystyle \psi (z)}
Digamma function . The special case ker
0
(
x
)
{\displaystyle _{0}(x)}
(
x
)
{\displaystyle (x)}
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 =x/{\sqrt {2}}+\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}.}
x )For integers n , kein x ) has the (complicated) series expansion
k
e
i
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
−
ln
(
x
2
)
b
e
i
n
(
x
)
−
π
4
b
e
r
n
(
x
)
+
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 \mathrm {kei} _{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}-\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\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}}
kei(x) for
x
{\displaystyle x}
k
e
i
(
x
)
e
x
/
2
{\displaystyle \mathrm {kei} (x)e^{x/{\sqrt {2}}}}
x
{\displaystyle x}
where
ψ
(
z
)
{\displaystyle \psi (z)}
Digamma function . The special case kei
0
(
x
)
{\displaystyle _{0}(x)}
(
x
)
{\displaystyle (x)}
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
{\displaystyle \beta }
f
2
(
x
)
{\displaystyle f_{2}(x)}
g
2
(
x
)
{\displaystyle g_{2}(x)}
(
x
)
{\displaystyle (x)}
See also
References
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
External links
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] 
![{\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}}](/media/api/rest_v1/media/math/render/svg/9d3b5d0a3ea65190b00e85e79f254bff6bc1acd8)
![{\displaystyle \mathrm {ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}(x/2)^{4k}}{[(2k)!]^{2}}}}](/media/api/rest_v1/media/math/render/svg/6709dbb4780c36d392283d858f0f84013cc1f81f)
![{\displaystyle \mathrm {ber} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha ]-{\frac {\mathrm {kei} (x)}{\pi }}}](/media/api/rest_v1/media/math/render/svg/065c7a5dffafdb6f53d39098519e283a42e17096)
![{\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}}](/media/api/rest_v1/media/math/render/svg/a2da961ae958ccb03f6a11e66ac59d36ff43314d)
![{\displaystyle \mathrm {bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}(x/2)^{4k+2}}{[(2k+1)!]^{2}}}}](/media/api/rest_v1/media/math/render/svg/f1603654dcdd94e66c03bea802c4404339fc87e9)
![{\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 }}}](/media/api/rest_v1/media/math/render/svg/6e2ec0b1655a880b04a2d93db587ca200e2484e0)
![{\displaystyle {\begin{aligned}\mathrm {ker} _{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}-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&{}\quad +{\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}}}](/media/api/rest_v1/media/math/render/svg/4f5436d43ed46ef81dc92ada4af6ad8d441e7f73)
![{\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}}](/media/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 ],}](/media/api/rest_v1/media/math/render/svg/6156e93597225fc773d61fff4b8c7b9b2ddc16f2)
![{\displaystyle \mathrm {kei} _{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}-\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\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}}](/media/api/rest_v1/media/math/render/svg/dca93bf8369c49807b90eb6ca6f51fc911ce153a)
![{\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}}](/media/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 ],}](/media/api/rest_v1/media/math/render/svg/7cb3fd63059e08690345a00b76f7425ba1bcc768)
