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 }(xe^{3\pi i/4}),\,}
where x is real, and
J
ν
(
z
)
{\displaystyle J_{\nu }(z)\,}
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
3
π
i
/
4
)
{\displaystyle K_{\nu }(xe^{3\pi i/4})\,}
K
ν
(
z
)
{\displaystyle K_{\nu }(z)\,}
th order modified Bessel function of the second kind.
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.
Ber(x) for
x
{\displaystyle x}
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
)
{\displaystyle \Gamma (z)}
Gamma function . The special case Ber
0
(
x
)
{\displaystyle _{0}(x)}
(
x
)
{\displaystyle (x)}
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}}
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
{\displaystyle n}
n
(
x
)
{\displaystyle _{n}(x)}
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
)
{\displaystyle \Gamma (z)}
Gamma function . The special case Bei
0
(
x
)
{\displaystyle _{0}(x)}
(
x
)
{\displaystyle (x)}
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)}
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 \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)+{\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}}
Ker(x) for
x
{\displaystyle x}
K
e
r
(
x
)
e
x
/
2
{\displaystyle \mathrm {Ker} (x)e^{x/{\sqrt {2}}}}
x
{\displaystyle x}
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}.}
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/f4a263b7ae69764facdf8d5e15fba1fc3c881552)
![{\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/f2a1665c8f8c8455c9690b5bd6a706fe45721959)
![{\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/3799488c90b2998fbe42342cbada195c0bd9f2d0)
![{\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/a46b67988e6164f1ba8d04c28d321e50540b7b10)
![{\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/897c4c42a2f29ca2704b4e0c306c7743d2281c5b)
![{\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/7ebcb459bc8a0a7c9f20c4cc42a5686bba13a091)
![{\displaystyle \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)+{\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}}](/media/api/rest_v1/media/math/render/svg/4fba166faf324a6356bbec96e1c5ec07a485be8a)
![{\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/fae7bb9569905c9119ac543e0878be564d56ca58)
![{\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/629b2c25ed4ccd9af090835f4e01f515b29dde4c)
![{\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/0660f8a0ffb80973f4d2ae7104400e9482b6b90e)
![{\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/59abd87da4b10eab7d8b3e0fece097c16b09055a)
![{\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/40522fc03a68410e4e99609659c881142b74855b)
