SYLLABUS

UNIT-1 :  Complex plane. Connected and Compact sets. Curves and Regions in complex plane. Jordan curve Theorem (statement only). Extended complex plane. Stereographic projection. Complex valued function Limits, Continuity and Differentiability. Analytic functions, Cauchy-Riemann equations (Cartesian and polar form). Harmonic functions, Construction of an analytic function.

UNIT-2 :     Complex integration, Complex line integrals, Cauchy integral theorem, Indefinite integral, Fundamental theorem of integral calculus for complex functions. Cauchy integral formula, Analyticity of the derivative of an analytic function, Morera's theorem, Poisson integral formula, Liouville' theorem.

UNIT-3    Taylor's theorem. Laurent's theorem. Maximum modulus theorem. Power series Absolute convergence, Abel's theorem, Cauchy-Hadamard theorem, Circle and Radius of convergence, Analyticity of the sum function of a power series.

UNIT-4 : Singularities of an analytic function, Branch point, Meromorphic and Entire functions, Riemann's theorem, Casorati-Weierstrass theorem. Residue at a singularity, Cauchy's residue theorem. Argument principle. Rouche's theorem.
Fundamental theorem of Algebra

UNIT-5 Conformal mapping. Bilinear transformation and its properties. Elementary
mappings: w (z) 1 2,e", sinz, cosz, and logz.
Evaluation of a real definite integral by contour integration. Analytic continuation. Power series method of analytic continuation.