Gamma Function of One Half/Proof 1

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Theorem

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


Proof

From the definition of the Beta function:

$\map \Beta {x, y} := \dfrac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$

Setting $x = y = \dfrac 1 2$:

\(\ds \map \Beta {\dfrac 1 2, \dfrac 1 2}\) \(=\) \(\ds \frac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2} } {\map \Gamma {\dfrac 1 2 + \dfrac 1 2} }\)
\(\ds \) \(=\) \(\ds \paren {\map \Gamma {\dfrac 1 2} }^2\)


Then from Beta Function of Half with Half:

$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$

Hence the result.

$\blacksquare$


Sources