Complex Analysis

Complex Plane, connected and compact sets. Extended complex plane: Stereographic projection. Complex valued functions, Analytic functions, C-R equation, Harmonic functions. Construction of an analytic functions

Complex Integration, Complex line Integrals, Cauchy’s integral theorem, Cauchy’s Fundamental theorem, Indefinite integrals, fundamental theorem of Integral Calculus for complex functions.

Cauchy’s Integral Formula, Analyticity of the derivative of an analytic function, Liouville’s theorem, Poisson’s Integral formula, Morera’s theorem, Maximum modulus principle. Taylor’s and Laurent’s Series.

Singularities. Meromorphic functions and Entire functions. Riemann’s theorem, Casorati-Weirstrass theorem. Rouche’s theorem. Fundamental theorem of algebra. Cauchy’s Residue theorem.

Contour integration. Conformal mapping (Introduction).Bilinear transformation and their simple properties . Power series.