Unit 2 : Complex Analysis

Limits, limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings.

Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability.

Unit 3

Analytic functions, examples of analytic functions, derivatives of functions, and definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy-Goursat theorem, Cauchy integral formula.

Unit 4

Liouville’s theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples.

Unit 5

Laurent series and its examples, absolute and uniform convergence of power series.