Introduction to Complex Numbers: complex plane, algebra of complex numbers; Functions of a Complex
Variable: mapping, limit, continuity, differentiability of a complex function, bilinear transformations; Analytic Functions: Cauchy-Riemann equations, necessary and sufficient conditions for differentiability; Harmonic Functions: Laplace equation in 2-dimensions in Cartesian and polar coordinates, relation with analytic function; Power Series: Taylor’s and Laurent’s expansion; Complex Integration: line integration over rectifiable curves, winding number, Cauchy’s Theorem: Cauchy’s integral formula, Liouville’s theorem, fundamental theorem of algebra; Residue Calculus: singularities, residues, Cauchy’s residue theorem, evaluation of integrals by contour integration, Rouche’s theorem; Conformal Mappings: preservation of angle, scale factors, applications; Meromorphic Functions: infinite product decomposition of entire functions; Schwarz-Christoffel Transformation: mapping real axis onto a polygon, triangles, and rectangles; Analytic Continuation: monodromy theorem.