Proposition: Gamma Function

The improper integral $\int_0^\infty \exp(-t)t^{x-1}dt$ is convergent if and only if $x > 0.$ For a given $x>0$, we call this limit the Gamma function $\Gamma(x)$ and set $$\Gamma(x):=\int_0^\infty \exp(-t)t^{x-1}dt,$$ where $\exp$ denotes the real exponential function. §§§1

Proofs: 1

  1. Proposition: Gamma Function Interpolates the Factorial

Proofs: 1
Propositions: 2

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  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