This text emphasizes the special functions that are used in complex analysis. Starting with the algebraic system of complex numbers, it offers an entry-level course on complex analysis of one variable. It presents the study of analytic functions, conformal mapping, analysis of singularities, and the computation of various integrals. The final three chapters introduce more advanced topics and applications. The book provides examples of applications to various physical problems and explains how to use Mathematica®, Maple™, and MATLAB®.
Table of Contents:
Complex number system
Algebraic structure
Triangle inequality and polar form
Topological properties
Stereographic projection
Analytic functions
Functions of a complex variable
Limit and continuity
Differentiability and analytic functions
Cauchy-Riemann equations
Applications to problems in potential flow
Harmonic functions
Power series and elementary functions
Sequences and series
Uniform convergence
Power series
Elementary functions
Multiple valued functions
Complex integration
Rectifiable arcs and contours
Line integral
Cauchy fundamental theorem and its implications
Cauchy theorem in various forms
Cauchy integral formula and its consequences
Morera theorem
Fundamental theorem of Algebra
Special functions
Gamma functions
Hypergeometric functions
Riemann zeta functions
Conformal mapping
Maximum modulus principle
Schwarz lemma
Subordination
Definition of conformal mapping
Bilinear transformation
Mappings related to discs and half-planes
Specific transformations such as 1/z, exp(z), sinz and zA2
Series representation and singularities
Taylor series and its consequences
Laurent series
Zeros and Poles of a function
Classification of residues
Meromorphic functions
Residue theorem and evaluation of integrals
Residue theorem
Argument principle and Rouche's theorem
Contour integration and applications
Improper integrals and evaluation of a real integral
Integrals involving sines and cosines
Integration through branch cut
Applications of conformal Mapping
Application to steady temperature
Application to electrostatic potential
Applications to fluid flow
Schwarz-Christoffel transformation
Dirichlet problem
Neumann problem
Harmonic functions
Harmonic functions and mean value property
Maximum principle
Poisson integral formula
Harnack Principle
Subharmonic and superharmonic functions
Topological properties of the class of analytic functions
Normal families
Arzela Ascoli theorem
Riemann Mapping theorem
Weierstrass Factorization theorem
Univalent function theory
Subclasses of univalent functions
Analytic continuation
Schwarz reflection principle
Entire functions and canonical products
Jensen's formula
Genus and order of an entire function
Hadamard theorem
Infinite products
Canonical product and Weirstrass theorem
Mittag Leffler's theorem
Elliptic functions
Doubly periodic function
Modular transformation
Properties of elliptic functions
Weierstrass elliptic functions
Related elliptic functions
Appendices
