# Math 406,  Spring 2006:  Overview of Lectures

 Week Lectures Asmar Topics covered 1 1-3 1.1, 1.2, 1.3 Origins of complex numbers; Real vs complex numbers; New operations: Re and Im, complex conjugate; The Complex Plane; Triangle inequality; Polar form; DeMoivre's Thm, generating trig IDs; Roots of unity. 2 4-5 1.3, 1.4, 1.5 Functions as maps; w and z planes; translation, dilation, rotation, inversion; exponential function; Euler ID. 3 6-7 1.5, 1.6 Functions: sin(z), cos(z); Trig IDs; nonuniqueness of sin, cos, exp; Hyperbolic trig fns. 4 8-10 1.6, 1.7 Flan 1.1-1.4 Hyperbolic trig fns; Log z; Arg vs arg; Roots & powers, principal value; x,y to z,zbar; Return to R^2: parameterized curves, line integrals, exact differentials; Div & Stokes Thms, Green's IDs & Thm. 5 11-13 3.1, 3.2 Contours & parameterizations; oriented curves; Complex line integrals; Differentiation in z, analytic fns, entire fns; Cauchy-Riemann eqs; harmonic fns, harmonic conjugate. 6 14-15 2.3, 2.4, 3.2, 3.4 Cauchy's Thm; Path independence; M-L inequality; more on entire functions; MIDTERM 1. 7 16-18 3.4, 3.6, 3.7, 4.1, 4.3 Continuous path deformation, Principle of path deformation; Cauchy's Integral Formula(s); fractional calculus; Integration around singularities; Mean Value Property; Max/Min Modulus Principle; Cauchy Estimates; Liouville's Thm. 8 19-21 4.1, 4.3, 4.4, 4.5 Power series; Absolute convergence, radius of convergence; Cauchy-Hadamard Thm; review of convergence tests, Gamma function; Taylor series; Laurent series, Laurent's Thm. - - - SPRING BREAK 9 22-24 3.6, 4.5, 4.6 Morera's Thm; Singularities & the Punctured Disk, Laurent series & Theorem; How to make Laurent Series; isolated zeros & singularities; Zeros of order N; Zeros-to-Poles Thm; Types of singularities: Poles, Essential, Removable; Principal part of a series; Residue; The Point at Infinity. 10 25-27 5.1, 5.2, 5.3, 5.4 (Flan.Ch7) Residue Thm; Useful formulae for calculating residues; Integrals of Trig Fns; Improper integrals; Semicircle and Box contours; Cauchy Principal Value. 11 28-30 5.4, 11.1 Improper integrals: convergence for semicircle with polynomials etc (ML inequality) and cos/sin (Jordan's Lemma); Intro to Fourier Transforms; Indented countours and Principal Values. 12 31-32 11.1 Fourier Transforms, inverse Fourier transform; Linearity; Fourier transform of real functions; ML inequality vs Jordan's Lemma; Review; MIDTERM 2. 13 33-35 11.2, 11.3 Square integrable fns; Fourier Transform (FT) of derivatives; Solving PDEs with FT; Shifting properties, Convolution Thm; Energies - Parseval & Plancherel Thms; Kernels; Distributions (generalized fns), delta distributions; Heat equation, heat kernel. 14 36-38 11.3, 11.4, 6.1, 6.2 More on solving PDEs with Fourier transforms; Overview of Laplace, Hilbert, & Hankel transforms; About Bessel functions; Special identities and the Shift property of Fourier transforms. Conformal Mappings; analytic functions & conformal maps; mapping isolines; rotation and dilation; Linear Fractional mappings; Lines-to-Circles Theorem. 15 39-41 6.1, 6.2, 6.3 Orthogonality of real and imaginary isolines; Joukowski map; Mapping with Log z; Mapping solutions of Laplace's eqs; Review for final.