5.2 Struve Functions
83
H 3
2
(z) =
z
2π
1 +
2
z 2
−
2
πz
sin(z) +
cos(z)
z
,
(5.3c)
H −
3
2
(z) =
2
πz
cos(z) −
sin(z)
z
,
(5.3d)
H (n+
1
2 ) (z) = Y (n+
1
2 ) (z) +
1
π
n
m=0
Γ
m +
1
2
z
2
−2m+n+
1
2
Γ (n + 1 + m)
and (5.3e)
H −(n+
1
2 ) (z) = (−1)
n J (n+
1
2 ) (z),
(5.3f)
with n = 0, 1, 2, · · · . In case no convergence could be reached the path integral
technique based on differential Eq. (5.1) will be used. For the initial condition we
will need the first derivative as well, which will be computed via
d
dz
H ν (z) = H ν−1 (z) −
ν
z
H ν (z).
(5.4)
Struve Function K
The Struve function K is defined via
K ν (z) = H ν (z) + Y ν (z),
(5.5)
and this equation is also the basis for evaluations. In case no sufficient convergence
could be reached, the path integral method will be used. The necessary derivative
could be calculated via
d
dz
K ν (z) = K ν−1 (z) −
ν
z
K ν (z).
(5.6)
5.2.2 Modified Struve Functions L and M
The modified Struve differential equation is given by
z
2 d 2 w
dz 2 + z
dw
dz
− (z
2
+ ν
2 ) = 4
1
2 z
ν+1
√
πΓ (ν +
1
2 )
,
(5.7)
with solution L ν (z) and M ν (z), and ν the order of the modified Struve functions.
For ν negative half-integers Eq. (5.7) becomes the modified Bessel differential
equation (4.29).