This course provides advanced topics in complex analysis  such as conformal mappings, and physical applications of conformal mapping.  Further topics include analytic continuation and Riemann surfaces, Rouches’s theorem and principle of the argument, mapping properties of analytic functions, inverse function theory, maximum modulus theory,  infinite products and gamma function, Sterling’s formula and Bessel’s functions, homotopy and homology theory, and analytic properties of holomorphic functions.