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Intermediate Methods of Mathematical Physics

James J. Kelly
Department of Physics
University of Maryland
College Park, MD 20742

    For several years I taught an introductory graduate course entitled Methods of Mathematical Physics at the University of Maryland, listed as PHYS604 and normally taken in the first semester.  The primary topics include: theory of analytic functions, integral transforms, generalized functions, eigenfunction expansions, Green functions, and boundary-value problems. The course is designed to prepare students for advanced treatments of electromagnetic theory and quantum mechanics, but the methods and applications are more general.

     Although this is a fairly standard course taught in most major universities, I was not satisfied with the available textbooks.  In my insufficiently humble opinion, the most widely used textbook, that by Arfken and Weber, is more useful as a reference than as a text  -- it is very thick, containing much more material than can be covered in a single semester, and is poorly organized, constantly referring forward to later chapters or to problems in later chapters for results needed in the present discussion.  A more recent textbook, by Lea, has a more manageable length but is written at a somewhat lower level than I wanted for my course and relegates some of the most important topics, such as asymptotic approximations and Green functions, to optional appendices.  Most of the other recent books of which I am aware suffer from some mixture of similar defects in emphasis, level, length, or organization.  The classic text by Matthews and Walker is more to my liking but is out of print and somewhat out of date.  Therefore, I soon found that preparation of lecture notes for distribution to students was evolving into a textbook-writing project even though the limited market for yet another text on a standard topic probably did not justify the effort.  On the other hand, I had already undertaken a similar project for a course on statistical physics

    I was not able to avoid producing too much material either.  I chose to skip most of the chapter on Legendre and Bessel functions, assuming that graduate students already had some familiarity with them, and instead refer them to a summary of properties that are useful for the chapter on boundary-value problems.  Other instructors might choose to omit the chapter on dispersion theory instead because most of it will probably be covered in the subsequent course on electromagnetism, but I find that subject more interesting and more fun to discuss than special functions.  Reviewers requested a chapter on group theory, but I have never reached that chapter in one semester.  Perhaps it will be useful for a two-semester course or at institutions where the average student is already well-versed in analytic functions, allowing more rapid coverage of some of the early chapters.  I believe that it should be possible to cover most of the remaining material well in a single semester at any mid-level university.  I assume that the calculus of variations will be covered in a concurrent course on classical mechanics and that the students are already comfortable with linear algebra, differential equations, and vector calculus.  Probability theory, tensor analysis, and differential geometry are omitted.

    I chose to prepare my lecture notes using Mathematica because I  am very enamored of its facility for combining mathematical typesetting with symbolic manipulation, numerical computation, and graphics into notebook documents approaching publication quality.  However, because students must learn the mathematical techniques in this course, not just the syntax of a program, all derivations in these lecture notes are performed by hand with Mathematica serving primarily as a word processor.  There are a few places where tedious but simple and uninteresting manipulations are performed by Mathematica, but those will probably be replaced by more traditional text if this project is ever published.  The figures were also produced using Mathematica, but the code for the figures is usually removed from the text.  And, of course, I often checked my work using the program.  However, I also discovered a disturbingly large number of integrals that Mathematica evaluated incorrectly.  Some of those errors have been corrected in later versions, perhaps due to my error reports, but inevitably new errors emerged even for integrals evaluated correctly in earlier versions!  The lessons that students should learn from my experience is either caveat emptor (let the buyer beware) and trust but verify.  The student must understand the mathematics well enough to recognize probable errors (the smell test) and to check the results of any mathematical software.  Software is helpful, but no software is perfect!  The wetware between your ears must evaluate the results of the software.   

    The notebooks may be accessed using the hyperlinks below.  To view the files directly, you must configure your web browser to launch either Mathematica or MathReader for files with the nb extension.  MathReader is a free program that will display or print a Mathematica notebook, but it cannot evaluate or edit notebooks.  Alternatively, you may download files to your computer and launch the application manually.  The notebooks were written during  2001-2003 and now use version 5.2, but can be used in earlier versions also.  Detailed solutions are available to all problems and can be supplied upon request to qualified instructors as Mathematica notebooks.  These solutions have not been made public so that instructors can assign homework without the solutions being in general circulation.  Some of the files will be updated as these materials are prepared for possible publication.

Last updated: July 21, 2006


nb pdf  Analytic functions
Introduces the theory of analytic functions.
Revised July 21, 2006
nb pdf  Integration
Primarily contour integration and use of integral representations.  Also offers a few useful tricks.
Revised July 21, 2006
nb pdf  Asymptotic series
Methods of steepest descent, partial integration, or expansion of an integrand. 
Revised July 21, 2006
nb pdf  Generalized functions
Dirac delta function and related generalized functions.  Introduction to Green functions.
Revised July 21, 2006
nb pdf   Integral transforms
Use of integral transforms, primarily Fourier or Laplace, for solving differential equations. 
Revised July 21, 2006
nb pdf  Analytic continuation and dispersion relations
The relationship between causality and analyticity is used to investigate the response of physical systems to external fields. 
Revised July 21, 2006
 nb pdf  Sturm-Liouville theory
A  fairly general development of eigenfunction expansions for self-adjoint differential operators.
Revised July 21, 2006
nb pdf   Legendre and Bessel functions
Detailed presentation of the properties of Legendre and Bessel functions in preparation for boundary-value problems.
Revised July 21, 2006
nb pdf   Boundary-value problems
A wide variety of boundary-value problems involving partial-differential equations in separable coordinate systems.  Emphasis is given to construction of Green functions.
Revised July 21, 2006
nb pdf   Group Theory
Introduction to both finite and continuous groups.  The irreducible representations of the quantum mechanical rotation group are analyzed in detail.
Revised July 21, 2006

Additional resources

Essential Mathematica for Students of Science
Teaches the fundamentals of Mathematica using a tutorial approach that is suitable for self study.
Statistical Physics using Mathematica
A graduate course in statistical physics that uses Mathematica fairly extensively.


    Please notify me of any errors you find.  Comments and suggestions are welcome.  My e-mail address is