Complex Analysis and Special Functions with Mathematical Software Tools

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