Introduction to Complex Analysis

Part 1 The complex plane: complex numbers; open and closed sets in the complex plane; limits and continuity. Part 2 Holomorphic function and power series: complex power series; elementary functions. Part 3 Prelude to Cauchy's theorem: paths; integration along paths; connectedness and simple connectedness; properties of paths and contours. Part 4 Cauchy's theorem: Cauchy's theorem, level I and II; logarithms, argument and index; Cauchy's theorem revisited. Part 5 Consequences of Cauchy's theorem: Cauchy's formulae; power series representation; zeros of holomorphic functions; the maximum-modulus theorem. Part 6 Singularities and multifunctions: Laurent's theorem; singularities; meromorphic functions; multifunctions. Part 7 Cauchy's residue theorem: counting zeroes and poles; claculation of residues; estimation of integrals. Part 8 Applications of contour integration: improper and principal-values integrals; integrals involving functions with a finite number of poles and infinitely many poles; deductions from known integrals; integrals involving multifunctions; evaluation of definite integrals. Part 9 Fourier and Laplace tranforms: the Laplace tranform - basic properties and evaluation; the inversion of Laplace tranforms; the Fourier tranform; applications to differential equations; proofs of the Inversion and Convolution theorems. Part 10 Conformal mapping and harmonic functions: circles and lines revisited; conformal mapping; mobius tranformations; other mappings - powers, exponentials, and the Joukowski transformation; examples on building conformal mappings; holomorphic mappings - some theory; harmonic functions.