Gamma Function of Minus 1
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Theorem
- $\map \Gamma {-1}$ is not defined.
where $\Gamma$ denotes the Gamma function.
Proof
\(\ds \map \Gamma 0\) | \(=\) | \(\ds \paren {-1} \, \map \Gamma {-1}\) | Gamma Difference Equation | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \Gamma {-1}\) | \(=\) | \(\ds \dfrac {\map \Gamma 0} {-1}\) |
But from Gamma Function of Zero, $\map \Gamma 0$ is not defined.
Hence the result.
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $33 \ \text{(e)}$