This mathematical physics course will develop the use of complex analysis in physics. It will develop the subject from an application point of view, and discuss its applications in Fourier transforms, Laplace transforms, Green's functions, differential equations, special functions, etc. It will be useful for those wishing to pursue theoretical physics.
Review of complex algebra and the complex plane, complex differentiation, Cauchy Riemann equations, analytic functions.
Complex power series: Taylor and Laurent expansions. Polynomial, rational, exponential, trigonometric, log, gamma and beta functions.
Complex Integral Calculus: Contour integration, Cauchy-Goursat theorem, Cauchy residue formula, Residue theorem, evaluation of integrals using complex methods, Schwarz reflection principle, analytic continuation. Poles and meromorphic functions, essential singularities, branch cuts. Kramers-Kronig relation.
Additional topics (if time permits): Integral transforms: Fourier integral and transforms. Laplace, Mellin transforms etc. Green functions and solving differential equations using them - various physical examples.