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. |