Gamma Function and Stirling's Approximation: Mathematical Methods

This lecture covers the gamma function and its properties, including its definition and analytic continuation. The instructor explains how the gamma function relates to factorials and demonstrates its application in complex analysis. The lecture also introduces Stirling's approximation, which provides an asymptotic formula for large factorials. The instructor discusses the significance of the poles of the gamma function and how to compute residues at these poles. Additionally, the lecture emphasizes the saddle point method for estimating the gamma function for large values. The instructor illustrates the derivation of Stirling's series and its importance in various fields of physics. Throughout the lecture, the instructor encourages student participation and feedback, highlighting the interactive nature of the course. The lecture concludes with practical examples and exercises to reinforce the concepts discussed, ensuring that students grasp the mathematical methods essential for their studies in physics.