Module 1(10 hours) Complex Functions, Spherical representations of extended complex plane, Analytic functions, Cauchy Riemann equations, Harmonic functions, Branches of multiple-valued functions, Branch point and Branch cut.

Module 2(10 hours) Complex Integral, Cauchy Goursat theorem, Cauchy's integral formula, Simply connected domain, Independence of paths.

Module 3 (10 hours ) Power Series, Taylor series, Laurent series, zeros and singularities, Residue calculus, Cauchy residue theorem, Evaluation of definite integrals, Argument principle, Rouches theorem.

Module 4 (10 hours) Conformal mappings, Geometry of Mobius transformations, Open mapping theorem, Maximum modulus theorem, Schwarz’s lemma, Partial fractions and factorization, Entire Functions, Picard Theorem.

To provide an overview of what is complex analysis and why it is important to study. To motivate how one can use the theory of complex analysis for evaluating many real analysis problems comfortably.

To introduce analytic functions, complex integral, entire functions, conformal mappings, singularities, mapping theorems and their applications.

To provide the fundamental concepts of complex analysis and point out the differences between real and complex in each context

To motivate for higher studies in advance complex analysis.

CO1: Students will learn complex differentiation, multivalued functions, analytic and harmonic functions.
CO2: Students will learn to compute complex integration and evaluate the definite integral.
CO3: They will learn the singularities and zeros of complex functions and will be able to evaluate their residues.
CO4: They will learn the conformal maps and Mobius transformations.
CO5: Students will have the knowledge and skills to solve problems independently. They will have the ability to use techniques from complex analysis and apply them to diverse fields like PDEs, Real Analysis, Fourier Transforms, and Analytic Number Theory.

J. B. Conway, Functions of One Complex Variable, Springer , 1978

T. W. Gamelin, Complex Analysis, Springer , 2000

J.E. Marsden and M.J. Hoffman, Basic Complex Analysis, W H Freeman & Co , 1998

D.G. Zill and P.D. Shanahan, Complex Analysis, Jones and Bartlet Student Edition , 2003