Mathematical Physics

Course Description

This is a one-semester graduate course on mathematical physics, centered on complex analysis and the classical special functions. The course begins with the foundations of complex analysis—Cauchy–Riemann equations, contour integrals, Cauchy’s theorem, Laurent series, and the residue theorem—and proceeds to asymptotic methods including Laplace’s method, stationary phase, and the method of steepest descent. The middle portion treats the Gamma, Bessel, Legendre, Hermite, Laguerre, Chebyshev, and hypergeometric functions from a complex-analytic viewpoint: each family is introduced through its generating function or integral representation, with series expansions and differential equations emerging as consequences. The course concludes with the calculus of variations, covering the Euler–Lagrange equation, constraints, Noether’s theorem, and the Hamiltonian formulation.

Course Material

The course is based on the lecture notes Mathematical Physics: Complex Analysis and Special Functions, with Arfken, Weber, and Harris’ Mathematical Methods for Physicists, 7th edition as the primary supporting reference.

References

The following references are recommended:

[1] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
[2] M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
[3] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Cambridge University Press, 2006.
[4] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
[5] N. N. Lebedev, Special Functions and Their Applications, Dover, 1972.