Syllabus

Limit, Continuity and Differentiability, analytic function, Cauchy Riemann equations, Laplace equation, Conformal mapping, branch and branch point, Linear fractional transformations, complex integration, line integral in the complex plane, Cauchy integral theorem, Cauchy integral formula, Liouville's theorem, Morera's theorem, sequence, series, convergence test, power series, functions given by 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.

After completing the course, the students will gather knowledge of complex variables, understand the basic theory of complex analysis, and gain the knowledge to apply the fundamental results from complex analysis in modern mathematics and applied sciences. The students will have the knowledge and skills to solve problems independently.

James W. Brown, Complex Variables and Applications, Tata McGraw Hill , 1990

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

Erwin Kreyszig, Advanced Engineering Mathematics, Wiley

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