82
5 Struve Functions and Related Functions
5.2
Struve Functions
5.2.1 Struve Functions H and K
Struve’s differential equation reads
z
2 d 2 w
dz 2 + z
dw
dz
+ (z
2
− ν
2 ) = 4
1
2 z
ν+1
√
π Γ (ν +
1
2 )
,
(5.1)
with solution H ν (z) and K ν (z), where ν is the order of the Struve functions. For ν
negative half-integers, −
1
2 , −
3
2 , · · · , Γ (ν +
1
2 ) → ∞ and thus the Eq. (5.1) becomes
the Bessel differential equation (4.10).
Struve Function H
The Struve function H ν (z) could be computed [1] via
H ν (z) =
∞
m=0
(−1)
m
z
2
2m+ν+1
Γ (m +
3
2 )Γ (ν + m +
3
2 )
.
(5.2)
To optimize the computation the denominator could be evaluated via
t (0) = Γ
3
2
Γ
ν +
3
2
t (1) =
3
2
ν +
3
2
· t (0)
t (2) =
1 +
3
2
1 + ν +
3
2
· t (1) · · ·
t (n) = (n − 1 +
3
2
)(n − 1 + ν +
3
2
) · t (n − 1),
and every summand could be built by cumulative products. Before summing up,
we split the series in real and imaginary part and again in positive and negative
members. Sorting in ascending, respectively, descending order before summing up
each of the four sub-series helps to minimize cancellation errors based on the finite
precision of the individual numbers.
Some of the summands of Eq. (5.2) will vanish for negative half-integers, but the
computation will be simplified due to the following equations:
H 1
2
(z) =
2
πz
(1 − cos(z)) ,
(5.3a)
H −
1
2
(z) =
2
πz
sin(z),
(5.3b)