Mathematical Methods Lecture 23 of 34

November 22, 2011 by Davide Brenci

K.S. Narain , ICTP

This lesson starts with the recall that the Fourier transform of a good function is also a good function and then beings the study of the Fourier transforms of a Generalized function. It was demonstrated that if the sequence {fn(x)} defines a Generalized function G(x) then the sequence of Fourier transforms {Fn(x)}, where Fn(x) is the Fourier transform of the function fn(x) for all n, defines also a Generalized function G(x), that is called the Fourier transform of G(x) and the inverse Fourier transform of G(x) give back G(x).

The particular case of the Dirac delta function was then studied in more details and some of it Properties demonstrated with the help of the Generalized function definition. The second example studied was the Step function and the sequence of functions that defines it was introduced. It was shown that the derivative of the Step function is the Dirac delta function. Then the Fourier transform of the Dirac delta function ?(x) was calculated and it was shown that is simply some constant, threfore the Fourier transform of a constant is again the Dirac delta function. Fallowing it was obtained that ?(y(x))=?i?(x-xi)/(dy/dx|x=xi) where y(xi)=0.

In the second part of the lesson the differential equations were studied and as an example the differential equation for the Laguerre polynomials ( x ?2/?x2 Lvn + (-x+v+1) ?/?x Lvn + n Lvn= 0) was derived. Then the definition of a differential equation of nth order was given as F(U(x),dU/dx,.....,dnU/dxn)=0. The problem consists in to obtain the function U(x). To solve this problem a theorem was introduced it says that: if the function H(U(x0),dU/dxIx=xo,.....,dn-1U/dxn-1Ix=xo) satisfy the Lipschitz Condition then dnU/dxn=H(U(x0),dU/dxIx=xo,.....,dn-1U/dxn-1Ix=xo). This result implies that the solution of the differential equation of nth order exist. As can be seen the information of the value of the function U(x0) and its derivative to order n-1 for x=x0 have to be known. This condition is known as the boundary condition and the solution depends which boundary condition is choused. In general the resulting differential equation could be very complicated but in this course we focused just in linear differential equations. Finally the definition for homogeneous and inhomogeneos differential equation was given and as an example the solution of a one order differential equation was obtained.

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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