Gamma Function for Non-Negative Integer Argument
Jump to navigation
Jump to search
Theorem
The Gamma function satisfies:
- $\map \Gamma z = \dfrac {\map \Gamma {z + 1} } z$
for any $z$ which is not a nonpositive integer.
Proof
From Gamma Difference Equation:
- $\map \Gamma {z + 1} = z \, \map \Gamma z$
which is valid for all $z \notin \Z_{\le 0}$.
The result follows by dividing by $z$.
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {I}$. The Gamma function: $5$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.4$: The Gamma Function for $n < 0$