**MATH50001 Analysis 2 (Complex Analysis) 2021**

Ari Laptev

**Course information
**

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

Recommended Schedule:

January 18-22 - Lectures 2-3

January 25-29 - Lectures 4-5 + Problem sheet 1

February 1-5 - Lecture 6-7 + Problems sheet 2

February 8-12 - Lectures 8-9 + Problem sheet 3

February 15-19 - Lectures 10-11 + Problem sheet 4

February 22-26 - Lecture 12

March 1-5 - Lectures 13-14 + Problem sheet 5

March 8-12 - Lectures 15-16 + Problem sheet 6

March 15-19 - Lectures 17-18 + Problem sheet 7

March 22-26 Lectures 19-20

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)

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.