**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.

Solutions CW1

Solutions

Solutions CW2

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)

Lecture 17 (pdf)

Lecture 18 (pdf)

Lecture 19 (pdf)

Lecture 20 (pdf)

All Lectures (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.