| Week | Lectures | Kreyszig | Topics covered |
| 1 | 1-3 | 13.1 - 13.3 |
Introducing complex numbers; New operations - complex conjugate, modulus, Re and Im; Polar form; Arg; Roots; De Moivre; Domains, functions. |
| 2 | 4-5 | 13.3, 13.4 | Derivatives, analytic, Cauchy-Riemann; harmonic functions, harmonic conjugate. |
| 3 | 6-7 | 13.4 - 13.7 |
More on harmonic conjugates; Exponential function, fundamental region; sin(z) and cos(z); Euler's formula; hyperbolic trig fns; Logarithm - ln(z) and Ln(z). |
| 4 | 8-10 | 13.7, 14.1, 14.2 |
More on logs, Principal value, branch cut; derivative of Ln(z); roots done with logs. Line integrals in the complex plane; simple, oriented curves; parameterized curves; Cauchy's Integral Theorem, piecewise smooth paths, path independence; ML inequality. |
| 5 | 11-13 | 14.2 - 14.4 |
Path independence; Principle of Path Deformation, multiply-connected paths; Cauchy Integral Formula, singularities; application to evaluating integrals; More on path manipulation; Integrals around multiple singularities. |
| 6 | 14-15 | 17.1 | Review; MIDTERM 1; Intro to Conformal Mapping; mapping curves. |
| 7 | 16-18 | 14.4, 17.1 |
ML and Cauchy inequalities; Liouville's Theorem; Morera's Thm; conformal maps, angle-preserving property; Simple maps; composition of maps. |
| 8 | 19-21 | 17.1 - 17.3 |
Zhukovsky map; mapping of curves; magnification, connection to Jacobian; Fractional Linear Transform (Mobius map); Lines & Circles Thm; Fixed Pt Thm, 3 Pt Thm; Fractional Calculus, Gamma function. |
| - | - | - | SPRING BREAK |
| 9 | 22-23 | 17.4, (17.5), 18.1 |
Mapping with trig fns; (Riemann surfaces); Applications to electrostatic potential probs; parallel plates, coaxial cylinders, wedges; solns to Laplace's eq. |
| 10 | 24-26 | 18.1, 18.2, 18.4 |
More on electrostatic problems; Mapping solutions with conformal maps; Orthogonality of isolines; Review of Mobius maps and Lines/Circles Thm; Applications to incompressible/irrotational fluid flow. |
| 11 | 27-29 | 18.4, 15.3, 15.4, 16.1 |
2D fluid flow problems, velocity potential and streamfunctions; streamlines; complex potential, complex velocity; flow past a cylinder; stagnation pts. Taylor Series for analytic fns; Taylor's Thm, Maclaurin series; Radius of convergence, uniqueness; Singularities, Laurent's Thm. |
| 12 | 30-31 | 15.4, 16.1 | Review for midterm; MIDTERM 2; Review of power series; Laurent series. |
| 13 | 32-34 | 16.1, 16.2, 15.3 |
Laurent's Thm, convergence in an annulus, punctured disk around singularity; uniqueness; How to find Laurent series; Isolated singularities: poles, essential; Principal part of L series; Poles to Zeroes Thm. Review series termwise manipulation. |
| 14 | 35-37 | 16.2, 16.3 |
Review singularities, developing Laurent series; Riemann sphere, analytic pt at infinity. Residue Method; definition, Residue Thm; Formulas. |
| 15 | 38-40 | 16.3, 16.4 |
Pole vs residue; Formulas for simple poles, pole order m; p/q Thm; residue integration of real integrals: improper integrals, trig integrals. |