**Table of Contents:**
- Foreword
- Complex numbers and functions of a complex variable
- Complex numbers (complex numbers; geometrical interpretation; stereographic projection; quaternions)
- Elementary transcendental functions
- Functions of a complex variable (complex functions of a real variable; functions of a complex variable; limits and continuity)
- Analytic and harmonic functions (the Cauchy-Riemann equations; harmonic functions; the geometrical meaning of the modulus and argument of a derivative)

- Conformal mappings connected with elementary functions
- Linear functions (linear functions; bilinear functions)
- Supplementary questions of the theory of linear transformations (canonical forms of linear transformations; some approximate formulae for linear transformations; mappings of simply connected domains; group properties of bilinear transformations ; linear transformations and non-Euclidean geometry)
- Rational and algebraic functions (some rational functions; mappings of circular lunes and domains with cuts; the function 1/2 (z + 1/z ); application of the principle of symmetry; the simplest non-schlicht mappings)
- Elementary transcendental functions (the fundamental transcendental functions; mappings leading to mappings of the strip and half-strip; the application of the symmetry principle; the simplest many-sheeted mappings)
- Boundaries of univalency, convexity and starlikeness

- Supplementary geometrical questions. Generalised analytic functions
- Some properties of domains and their boundaries. Mappings of domains
- Quasi-conformal mappings. Generalised analytic functions

- Integrals and power series
- The integration of functions of a complex variable
- Cauchy's integral theorem
- Cauchy's integral formula
- Numerical series
- Power series (determination of the radius of convergence; behaviour on the boundary; Abel's theorem)
- The Taylor series (the expansion of functions in Taylor series; generating functions of systems of polynomials; the solution of differential equations)
- Some applications of Cauchy's integral formula and power series (Cauchy's inequalities; area theorems for univalent functions; the maximum principle; zeros of analytic functions; the uniqueness theorem; the expression of an analytic function in terms of its real or imaginary part)

- Laurent series, singularities of single-valued functions. Integral functions
- Laurent series (the expansion of functions in Laurent series; some properties of univalent functions)
- Singular points of single-valued analytic functions (singular points; Picard's theorem; power series with singularities on the boundary of the circle of convergence)
- Integral functions (order; type; indicator function)

- Various series of functions. Parametric integrals. Infinite products
- Series of functions
- Dirichlet series
- Parametric integrals (convergence of integrals; Laplace's integral)
- Infinite products

- Residues and their applications
- The calculus of residues
- The evaluation of integrals (the direct application of the theorem of residues; definite integrals; integrals connected with the inversion of Laplace's integral; the asymptotic behaviour of integrals)
- The distribution of zeros. The inversion of series (Rouche's theorem; the argument principle; the inversion of series)
- Partial fraction and infinite product expansions. The summation of series

- Integrals of Cauchy type. The integral formulae of Poisson and Schwarz. Singular integrals
- Integrals of Cauchy type
- Some integral relations and double integrals
- Dirichlet's integral, harmonic functions, the logarithmic potential and Green's function
- Poisson's integral, Schwarz's formula, harmonic measure
- Some singular integrals

- Analytic continuation. Singularities of many-valued character. Riemann surfaces
- Analytic continuation
- Singularities of many-valued character. Riemann surfaces
- Some classes of analytic functions with non-isolated singularities

- Conformal mappings (continuation)
- The Schwarz-Christoffel formula
- Conformal mappings involving the use of elliptic functions

- Applications to mechanics and physics
- Applications to hydrodynamics
- Applications to electrostatics
- Applications to the plane problem of heat conduction

- Answers and Solutions