Physics 503
Methods of Mathematical Physics
Fall 2006

Tuesday & Thursday, 9:50-11:10am.
Tuesday, 5:40-7:00pm
Physics P-125

Final Exam is on Monday, December 18, 10:00am-12:30pm in room B-131

Prerequisites

Topics to be covered

• Functions of a complex variable (6 lectures)
  • Complex numbers
  • Analytic functions
• Applications of functions of a complex variable (9 lectures)
  • Contour integration
  • Conformal mapping
• Introduction to asymptotic methods (4 lectures)
  • Asymptotic series
  • Laplace method
  • Method of steepest descent
  • Method of stationary phase
• Special functions (6 lectures)
  • Gamma function
  • Bessel functions
  • Other special functions
• Basics of topology (3 lectures)
  • Topological spaces
  • Homotopy theory
  • Classification of textures and defects of ordered media

Homeworks

• Homework #1, due Thursday, September 21, 2006 (PDF )
  
  
• Homework #2, due Tuesday, October 3, 2006 (PDF )
  
  
• Homework #3, due Tuesday, October 17, 2006 (PDF )
  
  
• Homework #4, due Tuesday, November 14, 2006 (PDF )
  
  
• Homework #5, due Thursday, November 30, 2006 (PDF )
  
  
• Homework #6, due Thursday, December 7, 2006 (PDF )
  
  
• Homework #7, due Thursday, December 14, 2006 (PDF )
  
  

Recommended Books

• G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a complex variable: Theorie and Technique, McGraw-Hill book company, New York, 1983.
  
  This is a very good textbook which contains most of the topics I am going to present.
  


• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York, 1999.
  
  This is a very good textbook on asymptotic methods.
  


• F. W. Byron, Jr. and R. W. Fuller, Mathematics of classical and quantum physics, Dover, New York, 1992.
  
  Good and affordable textbook on different mathematical methods.
  

Additional Books

• G. M. J. Ablowitz and A. S. Fokas, Complex variables. Introduction and Applications., Cambridge University Press, Second edition, 2003.
  
  This book is a very good book on complex analysis. Close in topics to Carrier, Krook and Pearson's book. Has an extensive chapter on Riemann-Hilbert problems.


• G. B. Arfken, H. J. Weber, Mathematical methods for physicists, Academic press, London, 2001.
  
  This book is probably the bestseller on the subject. It is not a good textbook but contains all of the topics (except for topology) and is nice to have as a reference.


• B. A. Fuchs and B. V. Shabat, Functions of a complex variable and some of their applications,v. I, Pergamon press, 1964.
  
  This book contains a lot of examples, especially on conformal mappings and residue calculus.


• B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov, Modern Geometry-Methods and Applications : Part II, the Geometry and Topology of Manifolds, (Graduate Texts in Mathematics, Vol 104) Springer Verlag, 1985, ISBN: 0387961623.
  
  This is a very nice books with the basics of homotopy theory and a lot of examples.


• Mikio Nakahara, Geometry, Topology, and Physics,3rd edition, Cambridge, Massachusetts, MIT press, 1987, ISBN: 0852740956.
  
  Many topics relating geometry and physics are presented. In particular, chapter 4 is dealing with homotopy groups and their use for physics.


• N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys., 51, 591 (1979). (PDF, subscription required )
  
  This is a review article which explains how to use the homotopy theory for the classification of topological defects in ordered media.

Some of the homotopy groups used in physics (PDF )

Syllabus of the course