Mathematical methods Lecture 22 of 34

November 6, 2012 by Multimedia Publications and Printing Services

Lecturer: K.S. Narain , ICTP

This lesson starts by the introduction of Generalized function and its applications in physics, an example is the Dirac delta function, that have a direct application in physics, for example in the point particle limit the Charge density is a Dirac delta function. Some properties of the delta function are used. Nevertheless this delta function is not continuous and is better to work with continuous functions for this reason the Generalized functions  were introduced to better define the delta functions. Then the Dirac delta function is defined as a limiting case of sequence of Gaussian functions of the form Dn(x) = Sqrt[n/pi]Exp[-nx2] the integral of this function is always one independently of the value of n and in the limit n goes to infinity it converge to the Dirac delta function this was demonstrated in the second part of the lesson.

To define in a rigorous way the Generalized function the concept of good functions was introduced: A good function is a function that is differentiable any number of time and converge to zero when Abs[x] goes to infinity faster that any power of 1/ Abs[x]n, an example is the function Exp[-ax2] multiplied for any polynomial as well as its derivatives. The definition for a fairly good functions was also given: A fairly good function is a function that is differentiable any number of time and converge to zero when Abs[x] goes to infinity as Abs[x]n for some finite value of n. If was noticed that if a function is a good function then its Fourier transform is also a good function.

Then a rigorous definition for the Generalized functions was given: Suppose a sequence of good functions {fn(x)} if the integral of any fn(x) multiplied by any good function g(x) converge for n goes to infinity then the sequence {fn(x)} defines a Generalized function G(x). It was shown that any good function can be always written as a Generalized function. As examples of Generalized functions it were presented the Step function and the constant G(x)=1. Any fairly good function p(x) can be considered a Generalized function and the sequence of good functions that generate it is fn(x)= p(x) Exp[-x2/n2].

In the second part of this lesson attention was given to the properties of the Generalized functions. It was obtained that the linear combination of generalize functions is also a generalize function. The multiplication between a fairly good functions and a generalize function was defined and it was shown to give as a result also a generalize function. Note that the product of two generalize functions it is not defined. The derivative of a generalize function was also defined as the sequence of the derivative of the good functions {dfn(x)/dx}.

Mathematics for Physicists (Text Book from Google e-books preview)

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

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