This book is the first and second part of the textbook "Theory of functions of a complex variable," which will be published as part of the author's cycle of textbooks but for sections of higher mathematics. The content of the textbook fully covers the program on the course of the theory of functions of a complex variable for technical universities with in-depth study of mathematics. The textbook presents a fairly detailed presentation of the theory, accompanied by a large number of disassembled examples and problems. The first part of the textbook contains theoretical material on the topics "Complex numbers," "Functions of a complex variable," "Derivative of a function of a complex variable," "Conformal mappings," "Integral of a function of a complex variable." Meets the current requirements of the Federal State Educational Standard of Higher Education. For students of higher technical educational institutions. It can be used for self-training and advanced training.