# Math 406,  Spring 2005:  Overview of Lectures

 Week Lectures Fisher Topics covered 1 1-3 1.1, 1.2 History & review of complex numbers; Fundamental Thm of Algebra; Re and Im; Arg; DeMoivre's Thm; lines and circles. 2 4-6 1.2-1.5 Roots; domains: open, closed, connected, starlike, convex; sequences, convergence; Abs converg implies converg; functions; exp, log. 3 7-9 1.5, 1.6 Log & log; roots, powers; trig fns; Euler's formula; fns as mappings. Line Integrals. 4 10-12 1.6, 2.1 Contour integrals in C; oriented curves; Green's Thm, harmonic fns; analytic fns; path independence; Cauchy-Riemann; harmonic conjugate. 5 13-15 2.1, 2.1.1 Sums and products of analytic & harmonic fns; plotting analytic fns; Flows and fields: sourceless & divergence-free; electrostatic analogy; Isolines of real and imaginary parts. 6 16-17 2.3 Green's Theorem; Cauchy's Thm; path independence; Cauchy's formula; MIDTERM 1 7 18-20 2.2, 2.3 Principle of path deformation; real trig integrals; power series; radius of convergence; Cauchy-Hadamard; analyticity theorem. 8 21-23 2.4 Cauchy's theorem and power series; generalized derivatives; fractional calculus; isolated zeros; M-L inequality; Morera's Thm; Liouville's Thm; Cauchy estimates. - - - SPRING BREAK 9 24-26 - Singularities: removable, isolated (poles and essential); Laurent series; Laurent's Thm; solving ODE's with series; Principal part of a series; Taylor/Maclaurin series, Residue Thm. 10 27-29 - Residue Thm; Improper integrals - semicircle method; P/Q & Pole Thms; More real integrals: trig, even (0, infty), with log(x); polyanalytic fns. 11 30-31 3.3 Linear fractional trans; improper integrals, rectangular method. 12 32-33 3.3 Linear fractional transformations, Triples Thm, Line & Circle Thm; MIDTERM 2; Polyanalytic functions. 13 34-35 - Bianalytic and biharmonic functions; N-harmonic functions; Conformal mappings; using non-conformal points; basic mappings. 14 36-38 - Electrostatics: fundamental solns, special mappings, singularities; Fluid flow: streamfunction, Milne-Thompson Circle Thm. 15 39-41 4.2, 5.1 Zhukovsky map; airfoils; complex integration for Fourier transforms.