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.


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