74
4 Bessel and Airy Functions
Computational Aspects: Hankel Functions H
(1)
ν (z), H
(2)
ν (z)
For ν half-integer, the function values are evaluated via:
H
(1)
0.5 (z) =
2
πz
exp(iz)
i
,
H
(1)
−0.5 (z) =
2
πz
exp(iz) ,
(4.23)
H
(2)
0.5 (z) =
2
πz
exp(−iz)
−i
,
H
(2)
−0.5 (z) =
2
πz
exp(−iz) ,
(4.24)
and [2]
H
(1)
n−
1
2
=
2
πz
i
−n exp(iz)
n−1
k=0
(−1)
k (n + k − 1)!
k!(n − k − 1)!
1
(2iz) k ,
(4.25)
H
(1)
−n−
1
2
(z) = (−1)
n iH
(1)
n+
1
2
(z) ,
(4.26)
H
(2)
n−
1
2
=
2
πz
i
n exp(−iz)
n−1
k=0
(−1)
k (n + k − 1)!
k!(n − k − 1)!
1
(2iz) k , (4.27)
H
(2)
−n−
1
2
(z) = (−1)
n+1 iH
(2)
n+
1
2
(z) ,
(4.28)
with n positive integer.
For general orders ν we stick directly to the basic definitions (4.14, 4.15).
Because the Hankel functions are solutions of the Bessel differential equation (4.10),
we make use of the path integral method, as described above, in case of convergency
issues.
4.3.2 Modified Bessel Functions
The modified Bessel’s differential equation reads
z
2 d 2
dz 2 + z
d
dz
− (z
2
+ ν
2 )
φ = 0,
(4.29)
with ν complex. The solutions are given by I ν (z) and I −ν (z), respectively, K ν (z),
with
I ν (z) =
z
2
ν
∞
k=0
z
2
2k
k!Γ (ν + k + 1)
and
(4.30)
K ν (z) =
π
2
I −ν (z) − I ν (z)
sin(νπ)
.
(4.31)