PROBLEM BOOK
IN THE THEORY OF FUNCTIONS
VOLUME I

This book contains more than 300 elementary problems, together with 85 pages of detailed solutions at the end. Problems are arranged according to increasing difficulty, with difficult problems starred. Hints for solution are given, when necessary, while full, rigorous solutions are provided for all problems.

Problems are built upon:

1. Fundamental concepts: Numbers, points, point sets, paths, regions.

2. Infinite sequences and series: Limits, infinite series with constant terms, convergence properties of power series.

3. Functions of a complex variable: Limits of functions, continuity and differentiability, simple properties of the elementary functions.

4. Integral Theorems: Integration in the complex domain, Cauchy’s integral theorems and integral formulas.

5. Expansion in series: Series with variable terms, uniform convergence, expansion in power series, behavior of power series on the circle of convergence.

6. Conformal mapping. Linear functions, stereographic projection, simple non-linear mapping problems.

Full solutions to all problems: