### Mathematical Methods Lecture 20 of 34

November 10, 2011 by K.S. Narain

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 en=Exp[i?n] where n runs from minus infinity to infinity and that the functions en form an 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.

Mathematics for Physicists (Text Book from Google e-books preview)Written by<span class="addmd"> Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4</span>

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