**Course Content:**** **

- 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. (3 Lectures)
- 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.(6 Lectures).
- 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 (8 Lectures)
- Goursat’s theorem. Morera’s theorem; Uniform limits of holomorphic functions. Cauchy’s estimates; Liouville’s theorem. Fundamental theorem of Algebra. (5 Lectures)
- 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 (9 Lectures)
- Residue theorem. Argument principle with a discussion about winding numbers. Rouche’s theorem. (6 Lectures)
- Conformal mappings. Schwarz lemma. Holomorphic automorphisms of the disc. Fractional Linear transformations. (6 Lectures).

**Learning Outcomes:**Upon successful completion of the course, students will be able to

(i) write proofs of various consequences of the main theorems learnt in the course,

(ii) solve a wide variety of problems in complex analysis, having seen chains of reasoning that connect various basic facts already known to them during this course.

**Text books:**

- Serge Lang,
*Complex Analysis*, Springer, 2013, ISBN: 1475730837, 9781475730838 - J. Bruna, J. Cufi,
*Complex Analysis,*Hindustan Book Agency, 2015, ISBN:978-93-80250-73-1. - Gamelin T.,
*Complex Analysis*, Springer, 2013, ISBN: 0387216073, 9780387216072

**References :**

- Tristan Needham,
*Visual Complex Analysis*, Clarendon Press (1998), ISBN: 0198534469, 9780198534464 - Lars V. Ahlfors,
*Complex Analysis*, McGraw Hill Education; Third edition (2017), ISBN: 978-1259064821 - Elias M. Stein, Rami Shakarchi,
*Complex Analysis**, Volume 2 of Princeton lectures in analysis*, Princeton University Press (2010), ISBN: 1400831156, 9781400831159 - R. P. Boas (edited by H. P. Boas),
*Invitation to Complex Analysis*, (Mathematical Association of America Books), - J. B. Conway,
*Functions of One Complex Variable*, Springer (2012), ISBN: 1461599725, 9781461599722. - David C. Ullrich,
*Complex Made Simple*, - D. Sarason,
*Complex Function Theory,*Hindustan Book Agency (2012), ISBN: 9788185931845 - Donald E. Marshall, Complex Analysis, Cambridge University Press (2019), ISBN: 110713482X, 9781107134829
- Robert Everist Greene, Steven George Krantz,
*Function Theory of One Complex Variable,*American Mathematical Society (2006), ISBN: 0821839624, 9780821839621 - R. Remmert,
*Theory of Complex Functions*, Springer (2012), ISBN: 1461209390, 9781461209393

** **