The field of complex numbers. Topology of the complex plane. The extended complex plane and Stereographic projection. Complex differentiability and the Cauchy-Riemann equations; angles under complex-differentiable maps.
Examples of complex-differentiable functions: functions defined by a power series; radius of convergence. The exponential and logarithm with the source of its multivaluedness - the argument function. The principal branch of the logarithm.
Complex line integrals. Cauchy’s integral theorem and existence of local primitive for a complex differentiable function on discs or convex domains. The homotopy form of Cauchy’s integral theorem. The Local Cauchy integral formula and the equivalence of complex differentiability and complex analyticity.
Goursat’s theorem. Morera’s theorem; Uniform limits of holomorphic functions. Cauchy’s estimates; Liouville’s theorem. Fundamental theorem of Algebra.
Inverse and Open mapping theorems; the local maximum modulus principle. Isolated singularities: removable, pole and essential. Riemann’s removable singularities theorem. Meromorphic functions; functions holomorphic at infinity. Laurent series expansions. Casoratti-Weierstrass theorem.
Residue theorem with applications to computations of Fourier transforms. Argument principle with a discussion about winding numbers. Rouche’s theorem. Hurwitz’s theorem on the limit of a sequence of an injective holomorphic mappings.
Conformal mappings: Schwarz lemma. Holomorphic automorphic automorphisms of the disc. Fractional Linear transformations. Riemann mapping theorem (without proof); various equivalent characterizations of simply connected planar domains. Analytic continuation: definition and examples; Schwarz reflection principle.
Holomorphic functions as vector fields and the physical interpretation of Cauchy-Riemann equations. Harmonic functions; harmonic conjugates and connection with holomorphic functions. Poisson integral formula for the disc as a consequence of the Cauchy integral formula.
- Serge Lang, Complex Analysis, Springer, 2013
- J. Bruna, J. Cufi, Complex Analysis, Hindustan Book Agency, 2015
- Gamelin T., Complex Analysis, Springer, 2013,
- Tristan Needham, Visual Complex Analysis, Clarendon Press (1998)
- Lars V. Ahlfors, Complex Analysis, McGraw Hill Education; Third edition (2017)
- Elias M. Stein, Rami Shakarchi, Complex Analysis, Volume 2 of Princeton lectures in analysis, Princeton University Press (2010)
- R. P. Boas, Invitation to Complex Analysis, (Mathematical Association of America Books), American Mathematical Society, Revised Edition (2010)
- J. B. Conway, Functions of One Complex Variable, Springer (2012)
- David C. Ullrich, Complex Made Simple, American Mathematical Society (2008)
- D. Sarason, Complex Function Theory, Hindustan Book Agency (2012)
- Donald E. Marshall, Complex Analysis, Cambridge University Press (2019)
- Robert Everist Greene, Steven George Krantz, Function Theory of One Complex Variable, American Mathematical Society (2006)
- R. Remmert, Theory of Complex Functions, Springer (2012)