4.3 Bessel Functions and Related Functions
75
I ν (z) is called the modified Bessel function of first kind, and K ν (z) the modified
Bessel function of second kind. The first derivative of the Bessel function I ν is given
by
d
dz
I ν (z) = I ν−1 (z) −
ν
z
I ν (z) .
(4.32)
Additional recurrence equations are listed in [1, 2] for both functions.
Computational Aspects: Bessel Function I ν (z)
The computation techniques used to compute function values are pretty similar to
the one for J ν (z). Again ν could be complex, for half-integer orders the following
equations hold:
I 0.5 (z) =
2
πz
sinh(z), I −0.5 =
2
πz
cosh(z),
(4.33a)
I 1.5 (z) =
2
πz
−
sinh(z)
z
+ cosh(z)
,
(4.33b)
I −1.5 (z) = −
2
πz
− sinh(z) +
cosh(z)
z
,
(4.33c)
and for n + 1 a positive integer, the following finite series expansion [2]:
I ±(n+
1
2 ) =
1
√
2πz
exp(z)
n
k=0
(−1) k (n + k)!
k!(n − k)!(2z) k
±(−1)
n+1 exp(−z)
n
k=0
(n + k)!
k!(n − k)!(2z) k
.
(4.34)
To evaluate Eq. (4.34) the same techniques as described above will be used. In case
no convergency could be reached, the path integral method will be used.
In case none of the equations above could be used, we use directly the
series expansion (4.30). Because I −n (z) = I n (z) for n integer, we could easily
map negative integers onto positive ones, to avoid divergency. The techniques as
described for the Bessel function J ν are used as well to optimize the evaluation. In
case no convergency could be reached we use the path integral technique based on
differential equation (4.29).
Computational Aspects: Bessel Function K ν (z)
The Bessel function K ν holds
K −ν (z) = K ν (z),
(4.35)
75
I ν (z) is called the modified Bessel function of first kind, and K ν (z) the modified
Bessel function of second kind. The first derivative of the Bessel function I ν is given
by
d
dz
I ν (z) = I ν−1 (z) −
ν
z
I ν (z) .
(4.32)
Additional recurrence equations are listed in [1, 2] for both functions.
Computational Aspects: Bessel Function I ν (z)
The computation techniques used to compute function values are pretty similar to
the one for J ν (z). Again ν could be complex, for half-integer orders the following
equations hold:
I 0.5 (z) =
2
πz
sinh(z), I −0.5 =
2
πz
cosh(z),
(4.33a)
I 1.5 (z) =
2
πz
−
sinh(z)
z
+ cosh(z)
,
(4.33b)
I −1.5 (z) = −
2
πz
− sinh(z) +
cosh(z)
z
,
(4.33c)
and for n + 1 a positive integer, the following finite series expansion [2]:
I ±(n+
1
2 ) =
1
√
2πz
exp(z)
n
k=0
(−1) k (n + k)!
k!(n − k)!(2z) k
±(−1)
n+1 exp(−z)
n
k=0
(n + k)!
k!(n − k)!(2z) k
.
(4.34)
To evaluate Eq. (4.34) the same techniques as described above will be used. In case
no convergency could be reached, the path integral method will be used.
In case none of the equations above could be used, we use directly the
series expansion (4.30). Because I −n (z) = I n (z) for n integer, we could easily
map negative integers onto positive ones, to avoid divergency. The techniques as
described for the Bessel function J ν are used as well to optimize the evaluation. In
case no convergency could be reached we use the path integral technique based on
differential equation (4.29).
Computational Aspects: Bessel Function K ν (z)
The Bessel function K ν holds
K −ν (z) = K ν (z),
(4.35)