Zeta Explained #69: Deep Dive into Gamma (Weierstrass Product Formula)

This is the 69th video in a series explaining the Riemann zeta function. The idea of the series is to start with basics and eventually work our way to advanced topics. Today, we follow the text "Introductory Complex Analysis" by Richard A. Silverman. This particular video covers some basic properties of factorial and Gamma, leading up to the Weierstrass product formula for 1/Gamma. We use a "modern" approach invoking the Hadamard factorization theorem. 00:00 - Intro 01:02 - Factorials and Euler's Integral Formula 09:36 - Rough Stirling Approximation 12:06 - Euler's Limit Formula 16:55 - Poles and Zeros 20:44 - Fake Gamma Functions 29:34 - Weierstrass and Hadamard Factorization Theorems 42:35 - Outro