The Gamma Function

Complex Analysis · Axiom Academy

One smooth curve through every factorial — and the analytic continuation that gives it poles across the whole plane. 1. The Recurrence That Builds the Factorial Integration by parts on the defining integral (with u=t^ z , dv=e^ -t dt ; the boundary term vanishes) gives the one identity that powers everything else: the functional equation — step up by one input, multiply by z seed , then iterate: the factorial falls out The recurrence ladders between integers, but Gamma is defined everywhere in the right half-plane. The cleanest non-integer value comes from the substitution t=u^ 2 ( ), which turns the Euler integrand into a Gaussian: Climbing the half-integer ladder Feed back through the recurrence: and . Every half-integer value is a rational multiple of . 3. Continuing Past the Integral: The Poles The integral only converges for . But the recurrence, read backwards , extends Gamma further left one strip at a time — this is analytic continuation : Each division by z pushes the domain one unit left. Iterating, covers all of except where a denominator factor is zero. At a factor in the denominator vanishes while the numerator , so has a simple pole at each non-positive integer. Writing pins down the residue: the residues alternate in sign and shrink like 1/n! — exactly what the curve does as it spikes left of the origin. You've built the Gamma function from the factorial up — and watched it spread across the whole complex plane.

This is the written version of the interactive lesson above. See the full Complex Analysis course.