These notes are a slightly expanded version of a one-semester course, PHYS 349, 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 any comments, criticisms, suggestions, or corrections.
The text is available either as a complete set of notes or as separate chapters, either as postscript or pdf, as detailed below. I've included the date on which the current version was last modified, and I'll try to keep the notes reasonably up to date. My next planned change is to fiddle with the order a little and extend the introduction to include sufficient acknowledgements.
Complete Set
(ps,
pdf)
(70 pages; 12/22/99)
0. Introduction : Syllabus and Class
Notes
(ps, pdf)
(5 pages; 12/22/99)
Summary and Outline; References; Mathematical Notation.
1. Analysis of Complex Functions
(ps,
pdf)
(17 pages; 12/22/99)
Complex plane; Differentiability; Cauchy-Riemann equations; Branches and cuts; Integration and Cauchy's theorem; Power series expansions; Singularities and Laurent Series; Calculus of residues and contour integration; Examples.
2.
Exact and Approximate Evaluation of Sums and Integrals
(ps,
pdf)
(10 pages; 12/22/99)
Asymptotic sequences and expansions; Stokes' phenomenon; Watson's lemma and Laplace's method; Riemann-Lebesgue lemma and method of stationary phase; Saddle point method.
3.
Solution of Ordinary Differential Equations
(ps,
pdf)
(11 pages; 12/22/99)
First order systems; Second order systems with constant coefficients; Reduction of order; Green's functions for initial value and boundary value problems; WKB method.
4.
Transform Calculus
(ps,
pdf)
(15 pages; 12/22/99)
Fourier transform; Shifting relations; Distribution theory; Convolution theorem; Rayleigh-Plancherel theorem; Parseval's theorem; Laplace transform; Initial and final value theorems; Browich inversion formula; Applications to differential equations; Propagators and the heat and diffusion equations.
5.
Sturm-Liouville Theory
(ps,
pdf)
(8 pages; 12/22/99)
Orthogonality and boundary conditions; Eigenvalues and eigenfunctions; Formal vector space view; Inhomogeneous equations and Green's functions; Self-Adjointness.
6.
The Calculus of Variations
(ps,
pdf)
(6 pages; 12/22/99)
Bernoulli's brachistochrone; Euler-Lagrange equations; First integrals; Fermat's principle of least time; Geodesics; Multivariate generalizations.
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