1 FOURIER SERIES ON THE CIRCLE 1.1 Motivation and Heuristics 1.1.1 Motivation from Physics 1.1.1.1 The Vibrating String 1.1.1.2 Heat Flow in Solids 1.1.2 Absolutely Convergent Trigonometric Series 1.1.3 *Examples of Factorial and Bessel Functions 1.1.4 Poisson Kernel Example 1.1.5 *Proof of Laplace’s Method 1.1.6 *Nonabsolutely Convergent Trigonometric&n