4.3 Bessel Functions and Related Functions
77
integer (z complex) we get
j n (z) =
π
2z
J n+
1
2
(z) = (−1)
n
π
2z
Y −n−
1
2
(z)
(4.40)
y n (z) =
π
2z
Y n+
1
2
(z) = (−1)
n+1
π
2z
J −n−
1
2
(z)
(4.41)
h
(1)
n (z) =
π
2z
H
(1)
n+
1
2
(z) = (−1)
n+1 i
π
2z
H
(1)
−n−
1
2
(z)
(4.42)
h
(2)
n (z) =
π
2z
H
(2)
n+
1
2
(z) = (−1)
n i
π
2z
H
(2)
−n−
1
2
(z)
(4.43)
i
(1)
n (z) =
π
2z
I n+
1
2
(4.44)
i
(2)
n (z) =
π
2z
I −n−
1
2
(4.45)
k n (z) =
π
2z
K n+
1
2
=
π
2z
K −n−
1
2
.
(4.46)
The computation will rely on the computation of the corresponding “capital
letter function.”
4.3.4 Kelvin Functions
After some coordinate gymnastics the heat equation becomes
d 2
dx 2 R(x) +
1
x
d
dx
R(x) − iR(x) = 0,
with solutions I 0 (i (1/2) x), K 0 (i (1/2) x). This motivates the introduction of the Kelvin
functions:
ber ν (x) + i · bei ν (x) = J ν
x exp
3πi
4
,
(4.47)
ker ν (x) + i · kei ν (x) =
1
2
πiH
(1)
ν
x exp
3πi
4
,
(4.48)
with ν ∈ R and x > 0 and real. (In most publications the Kelvin functions are
written without the function argument in brackets, e.g., ber ν x, and sometimes they
are called Thomson function.) The computation will be based on the corresponding
Bessel function and taking the real, respectively, imaginary part.