Trodden M. Methods of Mathematical Physics I - Lecture Notes

Department of Physics Case Western Reserve University, 1999. - 70 pages.This course introduces some of the mathematical techniques that are frequently used in the analysis of a variety of physical processes. These notes are a slightly expanded version of a one-semester course that I gave at Case Western Reserve University in the Fall of 1999. Although I chose several textbooks to accompany the course, much of the material is presented in the way that I was taught it. I learnt mathematical physics from many people at Cambridge University, but those whose lectures clearly influenced these notes are Alan Macfarlane, Stephen Siklos, and John Stewart. In places I have borrowed heavily from their presentations and so these notes are certainly not a wholly original effort. However, I have tried to synthesize the material in a useful and orderly way, and have contributed significant material where I felt that more clarity was needed. It is my hope that these notes will evolve into a better and better resource as time passes. Certainly there is much room for improvement, and I welcome comments and suggestions from anyone who reads them.
The basis for the techniques studied in this course is a firm understanding of functions of a complex variable and we shall begin by laying the groundwork for this. In particular, we will spend some time reviewing and gaining expertise in performing contour integration of complex functions.
The experience we build up with contour integration will be useful when we next turn to the evaluation of sums and integrals. We will investigate a number of techniques, but the focus of this part of the course will be the various methods of obtaining asymptotic expansions of integrals. While a full understanding of this technique is quite challenging, it forms the basis of many physical calculations across many subdisciplines; particle physics, statistical mechanics, condensed matter physics.
The third part of the course concerns ordinary differential equations (ODEs). We will use everything we have learned up to this point so it is important to keep up with the material as the course progresses. We will investigate some classic techniques such as Green's functions and the WKB approximation, as well as introducing some of the special functions of mathematical physics that will be explored more fully in Physics 350.
In the final section of the course I will introduce Transform Calculus. I will cover the Fourier and Laplace transforms in some detail, and will make a number of remarks about transforms involving general kernel functions.