Gamma and Pi

This is a recording of a talk given to the Mathematics Discord on Pi Day 2023. Abstract: Ever wonder what the factorial of 1/2 is? We'll introduce the Gamma function, an extension of the factorial. We'll explore the function a little, and then prove Stirling's famous approximation for n!. Along the way, we'll see some important tools in analysis and a few surprise appearances from pi. Prerequisites: Integral calculus. Multivariable calculus and real analysis could be helpful. Links to proofs mentioned: Bohr Mollerup theorem: –Mollerup_theorem Gamma function Reflection Formula: 's_Reflection_Formula Method of Steepest Descent: 's_method 0:00 Intro and Factorial 5:25 The Gamma Function 21:08 Gamma(1/2) 50:53 Stirling's Approximation - First Form 1:14:51 Method of Steepest Descent 1:42:58 Stirling's Approximation - Full Form