Mathematics

**MTH 407: Complex Analysis I (4)**

**Pre-requisites (Desirable)**: MTH 303: Real Analysis I

*Learning Objectives*:

The aim of this course is to introduce the theory of modular forms with minimal prerequisites. The learning objectives of this course include the definition of analyticity, the Cauchy-Riemann equations and the concept of differentiability. Also to be learnt are the theorems on entire functions, residue theorem and applications and finally conformal mapping.

*Course Contents*:

- Complex numbers: powers and roots, geometric representation, stereographic projection
- Complex differentiability: limits, continuity and differentiability, Cauchy Riemann equations, definition of a holomorphic function
- Elementary functions: sequences and series, complex exponential, trigonometric, and hyperbolic functions, the logarithm function, complex powers, Mobius transformations
- Complex integration: contour integrals, Cauchy's integral theorem in a disc, Cauchy’s Integral Formula, Liouville’s theorem, Fundamental Theorem of Algebra, Morera’s theorem, Schwarz reflection principle
- Series representation of analytic functions: Taylor series, power series, zeros and singularities, Laurent decomposition, open mapping theorem, Maximum Principle
- Residue theory: residue formula, calculation of certain improper integrals, Riemann’s theorem on removable singularities, Casorati Weierstrass theorem, the argument principle and Rouche's theorem
- Conformal mappings: conformal maps, Schwarz lemma and automorphisms of the disk and the upper half plane

*Suggested Books*:

**Texts**

- Elias M. Stein, Rami Shakarchi, Complex Analysis, Princeton University Press, 2003
- Theodore W. Gamelin, Complex Analysis, Springer Verlag, 2001
- John B. Conway, Functions of one Complex Variable I, Springer, 1978
- E. Freitag and R.Busam, Complex Analysis, Springer, 2005

**References**

- Lars Ahlfors, Complex Analysis. McGrawHill, 1979
- R. Remmert, Theory of Complex Functions. Springer Verlag, 1991
- C. Caratheodory, Theory of Functions of a complex variable, AMS Chelsea, 2001

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