The book of the prominent English mathematician E. Titchmarsh, written in the 30s, was first published in Russian in 1951. It can certainly be attributed to classical works, and it still has not lost its significance. The book contains a lot of material that is not included in our common textbooks. Its author - a brilliant analyst and teacher - perfectly sets out various topics of the analytical theory of functions, convexly shading the leading ideas of calculations. There are many examples and tasks in the book.