Integral representation of Struve function (3)

Theorem

The following formula holds for $\mathrm{Re}(\nu)>-\dfrac{1}{2}$ and $|\mathrm{arg}(z)|< \dfrac{\pi}{2}$: $$\mathbf{H}_{\nu}(z)=Y_{\nu}(z) + \dfrac{z^{\nu}}{2^{\nu-1}\sqrt{\pi}\Gamma(\nu+\frac{1}{2})} \displaystyle\int_0^{\infty} e^{-zt} (1+t^2)^{\nu-\frac{1}{2}} \mathrm{d}t,$$ where $\mathbf{H}_{\nu}$ denotes the Struve function, $Y_{\nu}$ denotes the Bessel function of the second kind, $\pi$ denotes pi, $\Gamma$ denotes the gamma function, and $e^{-zt}$ denotes the exponential function.

References

• 1964: {{ #if: |{{{2}}}|Milton Abramowitz}}{{#if: Irene A. Stegun|{{#if: |, {{ #if: |{{{2}}}|Irene A. Stegun}}{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and {{ #if: |{{{2}}}|Irene A. Stegun}}}}|}}: [[Book:Milton Abramowitz/Handbook of mathematical functions{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Handbook of mathematical functions{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}}|{{#if: | ({{{ed}}} ed.)}}}}]]{{#if: | (translated by [[Mathematician:{{{translated}}}|{{ #if: |{{{2}}}|{{{translated}}}}}]])}}{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Integral representation of Struve function (2) | ... (previous)|}}{{#if: Recurrence relation for Struve fuction | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: $12.1.8$