Integral representation of Struve function (2)

Theorem

The following formula holds for $\mathrm{Re}(\nu)>-\dfrac{1}{2}$: $$\mathbf{H}_{\nu}(z) = \dfrac{z^{\nu}}{2^{\nu-1}\sqrt{\pi}\Gamma(\nu+\frac{1}{2})} \displaystyle\int_0^{\frac{\pi}{2}} \sin(z \cos(\theta)) \sin^{2\nu}(\theta) \mathrm{d}\theta,$$ where $\mathbf{H}$ denotes the Struve function, $\pi$ denotes pi, $\Gamma$ denotes gamma$, $\sin$ denotes sine, and $\cos$ denotes cosine.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $12.1.7$