Difference between revisions of "Integral representation of Struve function (3)"
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The following formula holds for $\mathrm{Re}(\nu)>-\dfrac{1}{2}$ and $|\mathrm{arg}(z)|< \dfrac{\pi}{2}$: | The following formula holds for $\mathrm{Re}(\nu)>-\dfrac{1}{2}$ and $|\mathrm{arg}(z)|< \dfrac{\pi}{2}$: | ||
$$\mathbf{H}_{\nu}(z)=Y_{\nu}(z) + \dfrac{z^{\nu}}{2^{\nu-1}\sqrt{\pi}\Gamma(\nu+\frac{1}{2})} \displaystyle\int_0^{\infty} e^{-zt} (1+t^2)^{\nu-\frac{1}{2}} \mathrm{d}t,$$ | $$\mathbf{H}_{\nu}(z)=Y_{\nu}(z) + \dfrac{z^{\nu}}{2^{\nu-1}\sqrt{\pi}\Gamma(\nu+\frac{1}{2})} \displaystyle\int_0^{\infty} e^{-zt} (1+t^2)^{\nu-\frac{1}{2}} \mathrm{d}t,$$ | ||
| − | where $\mathbf{H}_{\nu}$ denotes the [[Struve function]], $Y_{\nu}$ denotes the [[Bessel Y|Bessel function of the second kind]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma] function, and $e^{-zt}$ denotes the [[exponential]] function. | + | where $\mathbf{H}_{\nu}$ denotes the [[Struve function]], $Y_{\nu}$ denotes the [[Bessel Y|Bessel function of the second kind]], $\pi$ denotes [[pi]], $\Gamma$ denotes the [[gamma]] function, and $e^{-zt}$ denotes the [[exponential]] function. |
==Proof== | ==Proof== | ||
Latest revision as of 19:53, 4 November 2017
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.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $12.1.8$
