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5 Struve Functions and Related Functions
Modified Struve Function L
The modified Struve function L ν (z) could be computed [1] via
L ν (z) =
∞
m=0
z
2
2m+ν+1
Γ (m +
3
2 )Γ (ν + m +
3
2 )
.
(5.8)
This equation is very similar to (5.2) and thus the practical computation follows the
same steps.
Half-integer orders are evaluated via
L 1
2
(z) =
2
πz
(cosh(z) − 1) ,
(5.9a)
L −
1
2
(z) =
2
πz
sinh(z),
(5.9b)
L 3
2
(z) = −
z
2π
1 −
2
z 2
−
2
πz
sinh(z) −
cosh(z)
z
,
(5.9c)
L −
3
2
(z) =
2
πz
cosh(z) −
sinh(z)
z
,
(5.9d)
L (n+
1
2 ) (z) = I −(n+
1
2 ) (z) −
1
π
n
m=0
(−1)
m Γ (2m + 1) 2 −2m z
2
−2m+n−
1
2
Γ (m + 1)Γ (n + 1 − m)
,
(5.9e)
L −(n+
1
2 ) (z) = I (n+
1
2 ) (z),
(5.9f)
with n = 0, 1, 2, · · · . In case of no convergence the path integral technique based
on differential Eq. (5.7) will be used. The first derivative is given by
d
dz
L ν (z) = L ν−1 (z) −
ν
z
L ν (z).
(5.10)
Modified Struve Function M
The modified Struve function M is defined by
M ν (z) = L ν (z) − I ν (z),
(5.11)
which serves also as basis for evaluations. In case no sufficient convergence could
be reached, the path integral method will be used. The necessary derivative for the
initial condition could be calculated via
d
dz
M ν (z) = M ν−1 (z) −
ν
z
M ν (z).
(5.12)