**MATH95007 Complex Analysis 2020**

Ari Laptev

**Course information
**

The main goal of this course is to give an introduction to basic facts of Theory Complex functions.

Office hour: Mondays 12:00-13:00

Content

- Complex numbers and the complex plane:Basic properties, Convergence, Sets in the complex plane.
- Functions on the complex plane: Continuous functions, Holomorphic functions. Power series. Integration along curves.
- Cauchy's Theorem and Its Applications: Goursat's theorem. Local existence of primitives and Cauchy's theorem in a disc. Evaluation of some integrals, Cauchy's integral formulae.
- Meromorphic Functions: Zeros and poles. Laurent's Theorem. The residue formula. The argument principle and applications.
- Conformal Mappings: Preservation of Angles, Mobius Transformations

Lectures

Lecture 1 (pdf), Lecture 2 (pdf), Lecture 3 (pdf), Lecture 4&5 (pdf), Lecture 6 (pdf),

Problems:

probl.2 ( sol.2 )

Courseworks:

The deadline for the second corsework is

**Recommended Student Texts:**

Barry Simon,
* A Comprehensive Course in Analysis,
Part 2A: Basic Complex Analysis, *
American Math Society, 2015.

Elias M. Stein & Rami Shakarchi,
* II Complex Analysis, *
Princeton University Press, 2003.

Elias M. Stein & Rami Shakarchi,
* I Fourier Analysis, *
Princeton University Press, 2003.

John M. Howie,
* Complex Analysis, *
Springer, 2007.

Walter Rudin,
*Real and Complex Analysis, *
2nd ed., McGraw-Hill, 1974.

Lars V. Ahlfors,
*Complex Analysis, *
3rd ed., McGraw-Hill, 1979.