Gamma Function for Non-Negative Integer Argument

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): $\S 16$: The Gamma Function for $n < 0$: $16.4$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 25$: The Gamma Function: The Gamma Function for $n < 0$: $25.4.$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.