Stirling's formula: Euler's integral and Gaussian integral

This lecture delves into Stirling's formula, which estimates the size of n factorial for large n by involving pi and e. The proof involves Euler's integral, where pi and e unexpectedly appear in a formula about permutations. By taking logarithms and making a quadratic approximation around the maximum, the lecture demonstrates how the formula is derived. The Gaussian integral, a famous integral, is also explored, showing how it relates to Stirling's formula through polar coordinates. The lecture concludes by explaining how the main terms of Stirling's formula, n to the n e to the minus n and the square root of 2 pi n, arise from specific contributions in the integral.