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30.3 Gamma

The gamma function, $\Gamma(x)$, is the generalization of the factorial function to continuous variables, defined so that for positive integers $n$, $$\Gamma(n+1) = n!$$ Generalizing to all positive numbers and non-integer negative numbers, $$\Gamma(x) = \int_0^{\infty} u^{x - 1} \exp(-u) \, du.$$