Difference between revisions of "Integral representation of Struve function"
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(Created page with "==Theorem== The following formula holds for $\mathrm{Re}(\nu) > - \dfrac{1}{2}$: $$\mathbf{H}_{\nu}(z) = \dfrac{z^{\nu}}{2^{\nu-1}\sqrt{\pi}\Gamma(\nu+\frac{1}{2})} \displayst...") |
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The following formula holds for $\mathrm{Re}(\nu) > - \dfrac{1}{2}$: | The following formula holds for $\mathrm{Re}(\nu) > - \dfrac{1}{2}$: | ||
$$\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,$$ | $$\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,$$ | ||
| − | where $\mathbf{H}_{\nu}$ denotes the [[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma] function, and $\sin$ denotes [[sine]]. | + | where $\mathbf{H}_{\nu}$ denotes the [[Struve function]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma]] function, and $\sin$ denotes [[sine]]. |
==Proof== | ==Proof== | ||
Latest revision as of 16:11, 4 November 2017
Theorem
The following formula holds for $\mathrm{Re}(\nu) > - \dfrac{1}{2}$: $$\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.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $12.1.6$
