Gamma Function of Minus 1

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: $33 \ \text{(e)}$