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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.

plot(gamma, (-5, 7), detect_poles=True).show(ymin=-70, ymax=70)

| | | | | created: 2020-01-25 20:43:25 | modified: 2020-02-16 06:58:31 | by: bookofproofs | references: [581], [586]

1.Proof: (related to "Gamma Function")

2.Proposition: Gamma Function Interpolates the Factorial

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

[586] Heuser Harro: “Lehrbuch der Analysis, Teil 1”, B.G. Teubner Stuttgart, 1994, 11. Auflage