| Lec. | Date | Chapter Section |
Topic |
|---|---|---|---|
| 1 | 1/8 | 1: 1.1 | Complex numbers and the complex plane. |
| 2 | 1/10 | 1: 1.2, 1.3 | Convergence. Sets in the complex plane. Compactness. |
| 3 | 1/12 | 1: 1.3, 2.1 | Regions. Continuous complex-valued functions. |
| - | 1/15 | - | Martin Luther King Day - no classes |
| 4 | 1/17 | 1: 2.2 | Complex differentiable functions. |
| 5 | 1/19 | 1: 2.2 | Complex differentiable functions. |
| 6 | 1/22 | 1: 2.3 | Series of complex numbers. Convergence of power series. |
| 7 | 1/24 | 1: 2.3 | Uniform convergence. Differentiation of power series. |
| 8 | 1/26 | 1: 2.3 | Complex exponential and trigonometric functions. |
| 9 | 1/29 | 1: 2.4 | Piecewise smooth curves. Integration along curves. |
| 10 | 1/31 | 1: 2.4 | Integration along curves. |
| 11 | 2/2 | 3: 6 | Complex logarithm and powers. |
| 12 | 2/5 | 2: 1 2: 2 |
Goursat's Theorem. Existence of a primitive and Cauchy Theorem in a convex region. |
| - | 2/7 | - | Classes cancelled due to winter storm. |
| 13 | 2/9 | 3: 5 | Integrals along homotopic curves. |
| 14 | 2/12 | 3: 5 | Simply connected regions and holomorphic functions in such regions. Cauchy Theorem for multiply connected regions. |
| 15 | 2/14 | Exam 1 Study guide pdf | |
| 16 | 2/16 | 2: 4 | Cauchy integral formulas. |
| 17 | 2/19 | 2: 4 | Cauchy inequalities. Liouville's Thm. Fundamental Thm of Algebra. |
| 18 | 2/21 | 2: 4 | Power series representation of holomorphic functions. The Identity Thm. |
| 19 | 2/23 | 2: 4 | Corollaries of the Identity Thm. Maximum modulus principle. |
| 20 | 2/26 | Re f, Im f, and harmonic functions. | |
| 21 | 2/28 | 2: 5 | Morera's Theorem. Sequences of holomorphic functions. |
| 22 | 2/30 | 2: 5 | Symmetry principle and Schwarz reflection principle. |
| 3/4-10 | Spring Break | ||
| 23 | 3/12 | 3: 1 | Singularities. Zeros and poles. |
| 24 | 3/14 | 3: 1, 2 | Residues at poles. Calculating residues. |
| 25 | 3/16 | 3: 2 | The Residue Formula. Contour integration and improper integrals. |
| 26 | 3/19 | 3: 2 | Contour integration and improper integrals. |
| 27 | 3/21 | 3: 2 | Contour integration and improper integrals. |
| 28 | 3/23 | Laurent series. | |
| 29 | 3/26 | Exam 2 Study guide pdf | |
| 30 | 3/28 | 3: 3 | More on singularities. Casorati-Weierstrass theorem. |
| 31 | 3/30 | 3: 3 | Riemann sphere. Singularities in the extended complex plane. |
| 32 | 4/2 | 3: 3 3: 4 |
Meromorphic functions in the extended complex plane. The Argument Principle. |
| 33 | 4/4 | 3: 4 | Rouché’s Theorem. Open Mapping Theorem. |
| 34 | 4/6 | 3: 6 |
Solving eg=f. Holomorphic functions and angles between curves. |
| 35 | 4/9 | 8: 1 | Conformal equivalence. |
| 36 | 4/11 | 8: 1 | Möbius (fractional linear) transformations. |
| 37 | 4/13 | 8: 1 | Conformal equivalence: more examples. |
| 38 | 4/17 | 8: 2 | Schwarz Lemma. Automorphisms of the unit disc. |
| 39 | 4/19 | 8: 2 | Automorphisms of the unit disc and upper half-plane. |
| 40 | 4/20 | 8: 3 | Riemann Mapping Theorem. |
| 41 | 4/23 | 6: 1 | The gamma function. |
| 42 | 4/25 | 6: 2, 7: 1 | Riemann zeta function - an overview. |
| 43 | 4/27 | Problems pdf |
Final Exam: Tuesday, May 1, 10:10 a.m. - noon in 106 McAllister. Study guide