### 8.33 Functional relations involving the gamma function

8.331

1. $\Gamma \left(x+1\right)=x\Gamma \left(x\right)$

2.

$\begin{array}{ll}\Gamma \left(x+a\right)\hfill & =\left(x+a-1\right)\Gamma \left(x+a-1\right)\hfill \\ =\frac{\Gamma \left(x+a+1\right)}{(x+a)}\hfill \end{array}$

3.

$\begin{array}{ll}\Gamma \left(x-a\right)\hfill & =\left(x-a-1\right)\Gamma \left(x-a-1\right)\hfill \\ =\frac{\Gamma \left(x-a+1\right)}{\left(x-a\right)}\hfill \end{array}$

8.332

1.

$\begin{array}{ll}|\Gamma \left(iy\right){|}^{2}=\frac{\pi}{y\mathrm{sinh}\pi y}\hfill & [y\text{is}\text{real}]\hfill \end{array}$

MO 3

2.

$\begin{array}{ll}{|\Gamma \left(\frac{1}{2}+iy\right)|}^{2}=\frac{\pi}{\mathrm{cosh}\pi y}\hfill & [y\text{is}\text{real}]\hfill \end{array}$

3.

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