Complex Analysis

Seminars on the theory of functions of a complex variable: from complex differentiability and holomorphic functions to conformal maps. The course successively introduces power series and elementary functions of a complex variable, Laurent series and isolated singular points, residues and their application to computing integrals, the argument principle and Rouché’s theorem, the maximum modulus principle and Schwarz’s lemma, concluding with the geometry of conformal maps.

Information about the course from open MIPT sources (semester, workload, department, exam syllabi, point-based grading scheme) is collected on the “Course passport” page.

Contents

First assignment

1. Complex differentiability. Holomorphic functions
2. Power series and elementary functions
3. Taylor series and the uniqueness theorem
4. Laurent series

Second assignment

5. Isolated singular points
6. Residues
7. Computing integrals via residues
8. The argument principle and Rouché’s theorem

Third assignment

9. The maximum modulus principle and Schwarz’s lemma
10. The geometric meaning of the modulus and argument of the derivative
11. Conformal maps
12. Conformal maps by elementary functions

About this guide

This guide was prepared based on notes from seminar sessions.

Author: M.A. Nedoshivin
Email: nedoshivin.ma@phystech.edu

Found a typo or an error? Email the address above.

The materials are distributed under the CC BY-NC-SA 4.0 license; see the “About the project” page for details.