Harmonic and subharmonic functions, Green’s function, and the Dirichlet problem
for the Laplacian; the Riemann mapping theorem (revisited) and characterizing
simple connectedness in the plane; Picard’s theorem; the inhomogeneous
Cauchy–Riemann equations and applications; covering spaces and the monodromy
Suggested books and references:
Narasimhan, R., Complex Analysis in One Variable, 1st ed. or 2nd ed. (with Y. Nievergelt), Birkhauser (2nd ed. is available in Indian reprint, 2004).
Greene, R.E. and Krantz, S.G., Functions Theory of One Complex Variable, 2nd ed., AMS 2002 (available in Indian reprint, 2009, 2011).