Gamma Function of One Half/Proof 5

Theorem

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


Proof

\(\ds \map \Gamma 1 \, \map \Gamma {\frac 1 2}\) \(=\) \(\ds 2^0 \sqrt \pi \ \map \Gamma 1\) Legendre's Duplication Formula
\(\ds \leadsto \ \ \) \(\ds 0! \, \map \Gamma {\frac 1 2}\) \(=\) \(\ds 0! \, \sqrt \pi\) Definition of Gamma Function
\(\ds \leadsto \ \ \) \(\ds \map \Gamma {\frac 1 2}\) \(=\) \(\ds \sqrt \pi\) Factorial of Zero

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.