The following proposition establishes a connection between the gamma function the factorial.

Proposition: Gamma Function Interpolates the Factorial

For all positive real numbers $x > 0$, the gamma function fulfills the functional equation $$\Gamma(x+1)=x\Gamma(x).$$ In particular, for all natural numbers it equals the factorial $n,$ formally $$\Gamma(n+1)=n!.$$

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
  2. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition