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

$\blacksquare$