MAT604
Complex Analysis
Dr. Megan Griffin-Pickering
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Topics covered:
| Week 1 (Lectures 1-2) | Algebraic and topological properties of the complex numbers. |
| Week 2 (Lectures 3-4) | Complex differentiation & the Cauchy-Riemann equations. |
| Week 3 (Lectures 5-6) | Applications of the Cauchy-Riemann equations. Harmonic functions. Complex Series. |
| Week 4 (Lectures 7-8) | Holomorphic functions defined as power series & examples. |
| Week 5 (Lectures 9-10) | Complex logarithm; branch cuts. Introduction to the complex path integral. |
| Week 6 (Lectures 11-12) | Estimation of path integrals. Cauchy's Theorem: Goursat's Lemma; Discs; Simply connected domains. |
| Week 7 (Lectures 13-14) | The Cauchy Integral Formula & Applications: Liouville's Theorem, Morera's Theorem. |
| Week 8 (Lectures 15-16) | Taylor series; Zeroes of holomorphic functions |
| Week 9 (Lecture 17) | The Maximum Modulus Principle; Definition of winding number |
| Week 10 (Lectures 18-19) | Classification of singularities; Laurent series. |
| Week 11 (Lectures 20-21) | The Residue Theorem; Contour Integration. |
| Week 12 (Lectures 22-23) | Contour Integration continued: Indentation and Jordan's Lemma. Functions of sine and cosine. Summation of Series. The Argument Principle. |
| Week 13 (Lectures 24-25) | Rouche's Theorem. Conformal mapping. Harmonic functions and the Dirichlet problem. |
| Week 14 (Lecture 26) | Further examples and Q&A |
Visualising winding number:
Here is a widget for plotting the image of the unit circle under a complex transformation: SingSurf