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PAPER V: COMPLEX ANALYSIS

Unit I
: Introduction to the concept of analytic function: Limits and continuity – Analytic functions – Polynomials – Rational functions – Conformality: Arcs and closed curves – Analytic functions in regions – Conformal Mapping – Length and Area – Linear Transformations: The Linear group – The Cross ratio – Elementary Riemann Surfaces.

Unit II
: Complex Integration: Line Integrals Rectifiable Arcs – Line Integrals as Functions of Arcs – Cauchy’s theorem for a rectangle - Cauchy’s theorem in a disk, Cauchy’s Integral formula: The Index of a point with respect to a closed curve – The Integral formula – Higher derivatives Removable singularities, Taylor’s Theorem – Zeros and Poles – The Local Mapping – The Maximum principle – chains and cycles.

Unit III
: The Calculus of Residues: The Residue theorem – The Argument principle – Evaluation of definite integrals. Harmonic functions: The Definitions and basic Properties – Mean value property – Poisson’s Formula – Schwarz’s Theorem.

Unit IV
: Series and Product Developments: Weierstrass Theorem – The Taylor Series – The Laurent Series – Partial fractions and Factorization: Partial Fractions – Infinite Products – Canonical Products – Gamma function – Striling’s Formula.

Unit V
: The Riemann Mapping Theorem – Statement and Proff – Boundary Behaviour – Use of the reflection principle – Analytic arcs – Conformal mapping of Polygons: The Behaviour at an angle – The Schwarz – Christoffel Formula – Mapping on a rectangle – General properties of elliptic functions – The Weirestrass r-function – The functions z(z) and s(z) – The Differential Equation.

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