Gamma Function Extends Factorial

Theorem

$\forall n \in \N: \map \Gamma {n + 1} = n!$

Proof

For $n = 0$:

$\ds \map \Gamma 1$

$=$

$\ds \int_0^\infty e^{-t} \rd t$

Definition of Gamma Function

$\ds $

$=$

$\ds \bigintlimits {-e^{-t} } 0 \infty$

$\ds $

$=$

$\ds 0 - \paren {-1}$

$\ds $

$=$

$\ds 1$

Then by Gamma Difference Equation:

$\forall z \in \Z_{> 0}: \map \Gamma {z + 1} = z \, \map \Gamma z$

Hence the result.

$\blacksquare$

Sources

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1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $28 \ \text{(b)}$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 16$: The Gamma Function: Recursion Formula: $16.3$
1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $139$.
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.4 \ (4)$
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1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): gamma function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gamma function
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 25$: The Gamma Function: Recursion Formula: $25.3.$