Gamma Function of One Half/Proof 5
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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 |
$\blacksquare$