Formula:KLS:01.02:12

<< Formula:KLS:01.02:11

formula in The gamma and beta function

The gamma and beta function >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{dt}{(r+\mathrm{i}t)^{\rho}(s-\mathrm{i}t)^{\sigma}}=\frac{(r+s)^{1-\rho-\sigma}\Gamma\left(\rho+\sigma-1\right)}{\Gamma\left(\rho\right)\Gamma\left(\sigma\right)}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 1.2 of KLS.

URL links

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<< Formula:KLS:01.02:11

formula in The gamma and beta function

The gamma and beta function >>