Functional equation

An identity, such as $\Gamma(z+1)=z\Gamma(z)$ , relating a function's values at related inputs.

A functional equation is an identity relating a function's values at systematically related inputs, rather than giving an explicit formula outright. The Gamma function's functional equation, $\Gamma(z+1)=z\Gamma(z)$ , both reproduces the factorial recursion $n!=n\cdot(n-1)!$ for positive integers and, run in reverse, is exactly the tool used to extend $\Gamma$ by analytic continuation to negative non-integer arguments where the defining integral no longer converges. The reflection formula, $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$ , is a second functional equation for $\Gamma$ , relating its values at $z$ and $1-z$ .

Examples

Applying $\Gamma(z+1)=z\Gamma(z)$ repeatedly recovers $\Gamma(n)=(n-1)!$ from $\Gamma(1)=1$ .
Running $\Gamma(z+1)=z\Gamma(z)$ backward as $\Gamma(z)=\Gamma(z+1)/z$ defines $\Gamma$ at $z=-1/2$ from its already-known value at $z=1/2$ .