Gamma Function of Minus One Half

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Theorem

$\map \Gamma {-\dfrac 1 2} = -2 \sqrt \pi$

where $\Gamma$ denotes the Gamma function.


Proof

\(\ds \map \Gamma {-\dfrac 1 2}\) \(=\) \(\ds \frac {\map \Gamma {\frac 1 2} } {-1/2}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds -2 \, \map \Gamma {\frac 1 2}\)
\(\ds \) \(=\) \(\ds -2 \sqrt \pi\) Gamma Function of One Half

$\blacksquare$


Sources