Module 1 (10 hours) Complex numbers, functions, algebra of complex numbers, limit, continuity, differentiability, analytic function, Cauchy Riemann equations, Laplace equation, Harmonic function.
Module 2 (8 hours) Branch point and branch cut, multivalued functions, Linear fractional transformations, Conformal mapping.
Module 3 (10 hours) Complex integration, line integral in the complex plane, Cauchy's theorem, Cauchy integral theorem, Cauchy integral formula, Liouville's theorem, Morera's theorem.
Module 4 ( 10) Sequence and series of complex functions, convergence test, power series, Taylor's, Maclaurin's and Laurent's series, uniform convergence, zeros, limit point of zeros, singularities, poles, residue theorem, evaluation of real integrals.

To provide an overview of the course using the tools complex variables and complex functions. To motivate how one can use the theory of complex analysis for evaluating many real analysis problems comfortably

To introduce analytic function, complex integral, and the calculus using complex functions.

To teach different techniques of complex variables for real application problems.

Solving theory and its applications to the problems.

CO1: Students will gather the knowledge of complex variables, complex functions, and multivalued functions.
CO2: They will learn to evaluate the complex integrals.
CO3: They will understand the Cauchy's theorem in different domains.
CO4: They will learn the singularities and zeros of complex functions and will be able to evaluate their residues.
CO5: They can compute the sum of complex series and real definite integrals. They will have the ability to use techniques from complex analysis and apply them to diverse fields in real-life problems.

James W. Brown, Ruel V. Churchil, Complex variables and applications, Tat McGraw Hill,1990

John H. Mathews, Russell W. Howell, Complex Analysis for Mathematics and Engineers, Jones and Bartlett

Erwin Kreyszig, Advanced Engeering Mathematics, Willey

Dennis G. Zill, Patrick D. Shanahan, A First Course in Complex Analysis with Applications, Jones and Bartlett