76
4 Bessel and Airy Functions
thus we do not have to worry about the sign of the order. For half-integer values
K ±0.5 (z) =
π
2z
exp(−z),
(4.36)
and [2]
K ±(n+
1
2 ) =
π
2z
exp(−z)
n
k=0
(n + k)!
k!(n − k)!(2z) k .
(4.37)
Because in many cases Eq. (4.31) does not converge sufficiently well, if necessary we will use [2]
K ν (z) =
πi
2
exp
π
2
νi
H
(1)
ν (iz).
(4.38)
4.3.3 Spherical Bessel Functions
The wave equation in spherical coordinates reads
∂ 2
∂r 2 +
2
r
∂
∂r
+
1
r 2 sin θ
∂
∂θ
sin θ +
1
r 2 sin
2 θ
∂ 2
∂φ 2 −
1
c 2
∂ 2
∂t 2
ψ(r, θ, φ, t) = 0 ,
and with the ansatz
ψ(r, θ, φ, t) = r
−1/2 R(r)Y m,n (θ, φ)exp(iωt)
we get the radial equation
d 2
dr 2 R(r) +
1
r
d
dr
R(r) +
ω 2
c 2 −
(n +
1
2 ) 2
r 2
R(r) = 0 ,
(4.39)
with solution
R = a · J n+
1
2
(ωr/c) + b · J −(n+
1
2 ) (ωr/c),
and this is rational behind the phrase spherical Bessel function.
It is common practice that for “ordinary” Bessel functions capital letters are used
and for the corresponding spherical Bessel function the same, but small letter. For n