Various equivalent definitions of holomorphic functions of one complex variable. Cauchy-Riemann equations and
Laplace equation. Harmonic conjugates of holomorphic functions. Dirichlet problem on discs and Poisson integral
representation of harmonic functions.
Cauchy integral theorem for simply connected domains. Cauchy’s integral formula. Taylor series expandability of
holomorphic and harmonic functions. Liouville’s theorem and Fundamental theorem of Algebra.
Local properties of holomorphic functions: inverse function theorem, open mapping theorem, maximum modulus
principle, the principle of analytic continuation. Convergence properties of sequences of holomorphic functions;
deduce Morera’s theorem as a corollary.
Laurent series expansions of holomorphic functions on Annulii, Laurent series and the classification of isolated
singularities. The connection between Laurent series expansion and Fourier series expansions of functions on the
circle-group. Argument principle and application to deducing Hurwitz’s theorem. Residue theorem and its
Normal families. Schwarz’s lemma. Riemann mapping theorem. Conformal Mappings. Mobius transformations as
conformal self-maps of the Riemann sphere. Analytic Continuation and the Monodromy theorem.
Runge’s theorem and various equivalent characterizations of simply connected domains in the plane.
Inhomogeneous Cauchy-Riemann equations and the Mittag-Leffler theorem.