Analysis II

The primary goal of this course is to give students the knowledge of methods of complex analysis and its relevant applications.

Enrollment into the current study year. For the exchange students, meeting of the course prerequisites will be checked by the Study committee of the school

1. COMPLEX NUMBERS
   Definitions and Algebraic Properties, Geometric meaning, Elementary Topology

2. DIFFERENTIATION
   Limits and Continuity, Differentiability and Holomorphicity, The Cauchy– Riemann Equations

3. EXAMPLES OF FUNCTIONS
   Infinity and the Cross Ratio, Exponential and Trigonometric Functions, The Logarithm and Complex Exponentials

4. INTEGRATION
   Definition and Basic Properties, Antiderivatives, Cauchy’s Theorem, Cauchy’s Integral Formula

5. CONSEQUENCES OF CAUCHY’S THEOREM

6. POWER SERIES
   Sequences and Completeness, Series, Sequences and Series of Functions, Regions of Convergence

7. TAYLOR AND LAURENT SERIES
   Power Series and Holomorphic Functions, Classification of Zeros and the Identity Principle, Laurent Series

8. ISOLATED SINGULARITIES AND THE RESIDUE THEOREM
   Classification of Singularities, Residues, Argument Principle and Rouché’s Theorem

Students will get familiar with the foundations of complex analysis and will learn how to apply residues to calculate the integrals of real variables.

• Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka, A First Course in Complex Analysis, (2002-2018). E-version
• R. Courant, Differential and Integral Calculus I. Blackie & Son Ltd (1961). E-version
• B. Demidovich et al, Problems in Mathematical Analysis, Mir Publishers (1972). E-version

• writen exam
• oral exam