Lecturer: K.S. Narain , ICTP

This lesson starts by the introduction of the **Fourier transform**** **formalism. It was demonstrated that any periodic continuous function f(?) within the interval –?< ? < ?, with first derivative also continuous, can be expressed as a linear combination of exponential functions of the form *e _{n}*=Exp[i?n] where n runs from minus infinity to infinity and that the functions

*e*form an

_{n}**Orthonormal basis**. These types of series are called

**Fourier series**. In the proof of the theorem it was needed the help of the

**Weierstrass theorem**,

**Cauchy's convergence test**for

**infinite series**and the

**Cauchy-Schwarz inequality.**

A new theorem was then introduced without prove that is: consider a function f(?) defined within the interval –?< ? < ? and that this function is of **Bounded variation** then the **Fourier series** converge to i) (f(?+0) + f(?-0))/2 if –?< ? < ? and to ii) (f(?) + f(-?))/2 if ?= ? or ?=- ?, note that f(?) does not have to be continue or periodic. This theorem was illustrated with some examples, in order to calculate the Fourier's coefficients to solve this examples the **Residue theorem** was used. To calculate the many of the sums involving integers the trick is to write some functions with give you poles about integers numbers and then you construct a function that give you the correct residues.

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4