# Analytic Functions

LEARNING MODULE DESCRIPTION

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**GENERAL INFORMATION**

1. Module title: Analytic Functions

2. Term: Winter

3. Duration:30 lectures+30 practical classes

4. ECTS: 6

5. Module lecturer: dr hab. Michał Jasiczak

6. E-mail: mjk@amu.edu.pl

7. Language: English

**DETAILED INFORMATION**

1. Module aim (aims)

The aim of the course is to present basic results of complex analysis. This includes relation between holomorphic and analytic functions, Cauchy's Theorems and Cauchy's Integral Formulae and properties of zeros of holomorphic functions. Such fundamental results as the Maximumum Modulus Principle or the Argument Principle will also be proved. Applications to real analysis will be discussed.

2. Pre-requisites in terms of knowledge, skills and social competences (where relevant): Basic knowledge of real analysis including differential calculus and the Riemann integral.

**READING
LIST**

◦
**J. B. Conway, Functions of One Complex
Variable I, Springer 1978.**

◦
**A. I. Markushevich, Theory of
Functions of a Complex Variable, American Mathematical Society 2011.**

◦
**W. Rudin, Real and Complex Analysis,
McGraw Hill Education 2005.**

**SYLLABUS:**

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Week 1: Complex numbers, analytic functions;

Week 2: The Cauchy- Hadamard formula, derivative, holomorphic functions;

Week 3: The Cauchy-Riemann equations, basic examples of holomorphic functions;

Week 4: Curves, integrals over curves, primitive function;

Week 5: Cauchy's integral formula for a disk and consequences;

Week 6: Zeros of holomorphic functions;

Week 7: Half-exam;

Week 8: The index of a closed curve.

Week 9: Cauchy's Integral Formula and Cauchy's Theorems;

Week 10: Cauchy's Integral Formula and Cauchy's Theorems;

Week 11: The Open Mapping Theorem, Goursat's Theorem;

Week 12: Singularities, Laurent series;

Week 13: Singularities, Laurent series;

Week 14: Half-exam;

Week 15: The Maximum Modulus Principle.