Integral representation of Struve function

Theorem

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $12.1.6$