Gamma Function of Minus One Half
Theorem
- $\map \Gamma {-\dfrac 1 2} = -2 \sqrt \pi$
where $\Gamma$ denotes the Gamma function.
Proof
| \(\ds \map \Gamma {-\dfrac 1 2}\) | \(=\) | \(\ds \frac {\map \Gamma {\frac 1 2} } {-1/2}\) | Gamma Difference Equation | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \, \map \Gamma {\frac 1 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \sqrt \pi\) | Gamma Function of One Half |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $33 \ \text{(a)}$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $9$