Nanoscale Group Postgraduate Lectures in Physics - The Theory of Distributions

Nanoscale Group Postgraduate Lectures in Physics

The Theory of Distributions

Nanoscale Group
University of Western Sydney

Introduction:
The rapid developments in Physics and Technology in the 20th century require a revision of the traditional tools of Classical Physics. The traditional methodology of modelling physical phenomena as boundary problems for a differential equation is not sufficient when applied to some problems in areas as Quantum Physics there was the need to introduce new concepts such as Linear Operators, Distributions, Complex Analysis and Computers.
In these Lectures I will present an introduction to one of these topics, The Theory of Distributions (also called The Theory of Generalized Functions). This theory generalises the function concept as the second name indicates.
A distribution is always differentiable and always has a Fourier Transform. Those properties make them a very powerful tool applicable to a wide range of problems. Wave Functions in quantum mechanics and Lasers Fields propagating in non-linear mediums are distributions. These and others examples will be discussed in these lectures.

Content:
1. Functionals and Basic Functions
2. Differentiation of Distributions
3. Multiplication of Distributions
4. Convolutions
5. Fourier Transforms of Distributions
6. Applications