Introduction to complex analysis/ H.A. Priestley.

1. The complex plane; Complex numbers; Algebra in the complex plane; Conjugation, modulus, and inequalities;

Exercises; 2. Geometry in the complex plane; Lines and circles; The extended complex plane and the Riemann sphere; Möbius transformations;

Exercises; 3. Topology and analysis in the complex plane; Open sets and closed sets in the complex plane; Convexity and connectedness; Limits and continuity;

Exercises; 4. Paths; Introducing curves and paths; Properties of paths and contours;

Exercises; 5. Holomorphic functions. Differentiation and the Cauchy-Riemann equations; Holomorphic functions;

Exercises; 6. Complex series and power series; Complex series; Power series; A proof of the Differentiation theorem for power series;

Exercises; 7. A cornucopia of holomorphic functions; The exponential function; Complex trigonometric and hyperbolic functions; Zeros and periodicity; Argument, logarithms, and powers; Holomorphic branches of some simple multifunctions;

Exercises; 8. Conformal mapping; Conformal mapping; Some standard conformal mappings; Mappings of regions by standard mappings; Building conformal mappings.

Exercises; 9. Multifunctions; Branch points and multibranches; Cuts and holomorphic branches;

Exercises; 10. Integration in the complex plane; Integration along paths; The Fundamental theorem of calculus;

Exercises; 11. Cauchy's theorem: basic track; Cauchy's theorem; Deformation; Logarithms again;

Exercises; 12. Cauchy's theorem: advanced track; Deformation and homotopy; Holomorphic functions in simply connected regions; Argument and index; Cauchy's theorem revisited;

Exercises; 13. Cauchy's formulae; Cauchy's integral formula; Higher-order derivatives;

Exercises. 14. Power series representation; Integration of series in general and power series in particular; Taylor's theorem; Multiplication of power series; A primer on uniform convergence; Exercises; 15. Zeros of holomorphic functions; Characterizing zeros; The Identity theorem and the Uniqueness theorem; Counting zeros;

Exercises; 16. Holomorphic functions: further theory; The Maximum modulus theorem; Holomorphic mappings;

Exercises; 17. Singularities; Laurent's theorem; Singularities; Meromorphic functions;

Exercises; 18. Cauchy's residue theorem; Residues and Cauchy's residue theorem. Calculation of residues;

Exercises; 19. A technical toolkit for contour integration; Evaluating real integrals by contour integration; Inequalities and limits; Estimation techniques; Improper and principal-value integrals;

Exercises; 20. Applications of contour integration; Integrals of rational functions; Integrals of other functions with a finite number of poles; Integrals involving functions with infinitely many poles; Integrals involving multifunctions; Evaluation of definite integrals: overview (basic track); Summation of series; Further techniques;

Exercises; 21. The Laplace transform.