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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 J ν (z ) , is the 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 K ν (z ) is the 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 , 0 ≤ φ <2 π . With the exception of bern (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 ) is the digamma function .
ber(x )
[ edit ]
ber(x ) for x between 0 and 20.
b
e
r
(
x
)
/
e
x
/
2
{\displaystyle \mathrm {ber} (x )/e^{x/{\sqrt {2}}}}
for 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 )
[ edit ]
bei(x ) for x between 0 and 20.
b
e
i
(
x
)
/
e
x
/
2
{\displaystyle \mathrm {bei} (x )/e^{x/{\sqrt {2}}}}
for 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 )}
, and
g
1
(
x
)
{\displaystyle g_{1}(x )}
are defined as for ber(x ).
ker(x )
[ edit ]
ker(x ) for x between 0 and 14.
k
e
r
(
x
)
e
x
/
2
{\displaystyle \mathrm {ker} (x )e^{x/{\sqrt {2}}}}
for 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 )
[ edit ]
kei(x ) for x between 0 and 14.
k
e
i
(
x
)
e
x
/
2
{\displaystyle \mathrm {kei} (x )e^{x/{\sqrt {2}}}}
for 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 ).
See also
[ edit ]
References
[ edit ]
Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 9" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 379. 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 , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .
External links
[ edit ]
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 ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Kelvin_functions&oldid=1188006858 "
C a t e g o r i e s :
● S p e c i a l h y p e r g e o m e t r i c f u n c t i o n s
● W i l l i a m T h o m s o n , 1 s t B a r o n K e l v i n
H i d d e n c a t e g o r i e s :
● A r t i c l e s l a c k i n g i n - t e x t c i t a t i o n s f r o m D e c e m b e r 2 0 2 3
● A l l a r t i c l e s l a c k i n g i n - t e x t c i t a t i o n s
● T h i s p a g e w a s l a s t e d i t e d o n 2 D e c e m b e r 2 0 2 3 , a t 2 0 : 3 4 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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