Mathematical Methods Lecture 21 of 34

November 15, 2011 by K.S. Narain

K.S. Narain , ICTP

This lesson focused on how to calculate the Fourier series for different examples and to demonstrate that the last theorem of the previous lesson holds. The first example was already considered in the previous lesson but this time for the solution it was used the trick involving the Residue theorem, explained in the last part of the previous lesson, to solve it. A second example considered was the calculus of the Fourier series for the function f(?)=abs(?) once more time the trick was used to recover the initial function.

In the second part of the lesson the Fourier series were extended to the Fourier transform but just doing a change of variables on ? that was not rigorous but it gave an idea of the relation between them. Then the Fourier transform was enounced and the condition, that the absolute value of the function has to be integrable, discussed. The Fourier inversion theorem was also studied. Then some examples were considered and the Fourier transform obtained, it was also shown that its reproduces all the values of the initial function. The first example was a continuous function and everything was well behaved, then in the last example a discontinuous function was considered.

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

Take some function that satisfy the conditions of the second theorem of lesson 20, write the Fourier series for it and then show that the theorem is satisfied.

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