Kelvin Functions1
The solution to the Bessel equation
x w xw ix w
22
0
′′ +′−=
, where
i
=
−1
, may be written
in terms of modified Bessel functions as
wx cI xi cK xi() () ()=+
10 2 0
The Kelvin functions,
ber( )x
,
bei( )x
,
ker( )x
and
kei( )x
, may be defined by their relations to
the modified Bessel functions
Ix
0
()
and
Kx
0
()
as given by
Ixi x i x
0() ()=+ber bei( )
and
Kxi x i x
0
() ()=+ker kei( )
Thus it follows that wx()
may be written as
wxc xixc xix() () ()=+
[]
++
[]
12
ber bei( ) ker kei( )
where the four Kelvin functions are real functions. These Kelvin functions may be evaluated
from the following infinite series:
ber( ) ()
[( )!]
xx
n
nn
n
n
=−
=
∞
∑1
22
4
42
0
(1)
bei( ) ()
[( )!]
xx
n
nn
n
n
=−
+
+
+
=
∞
∑
1
221
42
42 2
0
(2)
ker( ) ( ) ln ( ) ()
[( )!]
xx xxx
nm
nn
n
m
n
n
=−+
+−
==
∞
∑∑
πγ
42
1
22
1
4
42
1
2
1
bei ber
(3)
kei ber bei() () ln () ()
[( )!]
xxxxx
nm
nn
n
m
n
n
=− − +
−−
−
−
−=
−
=
∞∑∑
πγ
42
1
221
1
42
42 2
1
21
1
(4)
where
γ
is Euler’s constant given by Abramowitz and Stegun2 as
γ
= ++++ + −
=
→∞
lim ln( ) .
m
mm11
2
1
3
1
4
10 577215664901532860606512K
Abramowitz and Stegun tabulate the above four Kelvin functions for
05≤≤x
. These Kelvin
functions have the following characteristics:
ber( )x
:
ber( )0
= 1 and oscillates with increasing amplitude as
x
increases
bei( )x
:
bei( )0
= 0 and oscillates with increasing amplitude as
x
increases
ker( )x
:
ker( )0
=
∞
and oscillates with decreasing amplitude as
x
increases
kei( )x
:
kei( )0
=
−
π
/4
and oscillates with decreasing amplitude as
x
increases
1 This documentation of Kelvin functions and KelvinFunctions.LIB were provided by G. E. Myers, Department
of Mechanical Engineering, University of Wisconsin-Madison, 2003.
2 Abramowitz, M. and I. A. Stegun (eds.): Handbook of Mathematical Functions, Applied Mathematics Series
55, National Bureau of Standards, 1964. [Dover, 1965].
