Gamma Function of Minus 3 over 2
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Theorem
- $\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$
where $\Gamma$ denotes the Gamma function.
Proof
| \(\ds \map \Gamma {-\dfrac 1 2}\) | \(=\) | \(\ds -\dfrac 3 2 \map \Gamma {-\dfrac 3 2}\) | Gamma Difference Equation | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \Gamma {-\dfrac 3 2}\) | \(=\) | \(\ds -\dfrac 2 3 \map \Gamma {-\dfrac 1 2}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac 2 3 \paren {-2 \sqrt \pi}\) | Gamma Function of Minus One Half | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {4 \sqrt \pi} 3\) |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $33 \ \text{(b)}$