**MATH50001 Analysis II (Complex Analysis) 2022**

Ari Laptev

**Course information
**

The main goal of this course is to give an introduction to basic facts from Complex Analysis

Recommended Schedule:

January 17-21 - Lectures 3-4 + + Problem sheet 1

January 24-28 - Lectures 5-6 + Problem sheet 2

January 31- February 4 - Lecture 7-8 + Problems sheet 3

February 7-11 - Lectures 9-10 + Problem sheet 4

February 14-18 - Lectures 11-12 + Problem sheet 5

February 21-25 - Lecture 13-14

February 28 March 4 - Lectures 15-16 + Problem sheet 6

March 7-11 - Lectures 17-18 + Problem sheet 7

March 14-18 - Lectures 19-20 + Problem sheet 8

Content

- Holomorphic Functions: Definition using derivative, Cauchy-Riemann equations, Polynomials, Power series, Rational functions, Moebius transformations.
- Cauchy's Integral Formula: Complex integration along curves, Goursat's theorem, Local existence of primitives and Cauchy's theorem in a disc, Evaluation of some integrals, Homotopies and simply connected domains, Cauchy's integral formulas.
- Applications of Cauchy's integral formula: Morera's theorem, Sequences of holomorphic functions, Holomorphic functions defined in terms of integrals, Schwarz reflection principle. Meromorphic Functions: Zeros and poles. Laurent series. The residue formula, Singularities and meromorphic functions, The argument principle and applications, The complex logarithm.
- Harmonic functions: Definition, and basic properties, Maximum modulus principle. Conformal Mappings: Definitions, Preservation of Angles, Statement of the Riemann mapping theorem.

Lectures

Lecture 1 (pdf)

Lecture 2 (pdf)

Lecture 3 (pdf)

Lecture 4 (pdf)

Lecture 5 (pdf)

Lecture 6 (pdf)

Lecture 7 (pdf)

Lecture 8 (pdf)

Lecture 9 (pdf)

Lecture 10 (pdf)

Lecture 11 (pdf)

Lecture 12 (pdf)

Lecture 13 (pdf)

Lecture 14 (pdf)

Lecture 15 (pdf)

Lecture 16 (pdf)

Lecture 17 (pdf)

Lecture 18 (pdf)

Lecture 19 (pdf)

Lecture 20 (pdf)

Problems:

probl.2 ( sol.2 )

probl.3 ( sol.3 )

probl.4 ( sol.4 )

probl.5 ( sol.5 )

probl.6 ( sol.6 )

probl.7 ( sol.7 )

**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.