14.7 Struve Functions

The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 12 and (DLMF 11). The Struve Function \({\bf H}_{\nu}(z)\) is a particular solution of the differential equation: $$ z^2 {d^2 w \over dz^2} + z {dw \over dz} + (z^2-\nu^2)w = {{4\left({1\over 2} z\right)^{\nu+1}} \over \sqrt{\pi} \Gamma\left(\nu + {1\over 2}\right)} $$

which has the general soution $$ w = aJ_{\nu}(z) + bY_{\nu}(z) + {\bf H}_{\nu}(z) $$

Function: struve_h (v, z)

The Struve Function H of order \(\nu\) and argument z:

$$ {\bf H}_{\nu}(z) = \left({z\over 2}\right)^{\nu+1} \sum_{k=0}^{\infty} {(-1)^k\left({z\over 2}\right)^{2k} \over \Gamma\left(k + {3\over 2}\right) \Gamma\left(k + \nu + {3\over 2}\right)} $$

(A&S eqn 12.1.3) and (DLMF 11.2.E1).

When besselexpand is true, struve_h is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.

Function: struve_l (v, z)

The Modified Struve Function L of order \(\nu\) and argument z: $$ {\bf L}_{\nu}(z) = -ie^{-{i\nu\pi\over 2}} {\bf H}_{\nu}(iz) $$

(A&S eqn 12.2.1) and (DLMF 11.2.E2).

When besselexpand is true, struve_l is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.