Complex Analysis and Applications, Second Edition explains complex analysis for students of applied mathematics and engineering. Restructured and completely revised, this textbook first develops the theory of complex analysis, and then examines its geometrical interpretation and application to Dirichlet and Neumann boundary value problems.
A discussion of complex analysis now forms the first three chapters of the book, with a description of conformal mapping and its application to boundary value problems for the two-dimensional Laplace equation forming the final two chapters. This new structure enables students to study theory and applications separately, as needed.
In order to maintain brevity and clarity, the text limits the application of complex analysis to two-dimensional boundary value problems related to temperature distribution, fluid flow, and electrostatics. In each case, in order to show the relevance of complex analysis, each application is preceded by mathematical background that demonstrates how a real valued potential function and its related complex potential can be derived from the mathematics that describes the physical situation.
Analytic Functions
Review of Complex Numbers
Curves, Domains, and Regions
Analytic Functions
The Cauchy-Riemann Equations: Proof and Consequences
Elementary Functions
Complex Integration
Contours and Complex Integrals
The Cauchy Integral Theorem
Antiderivatives and Definite Integrals
The Cauchy Integral Formula
The Cauchy Integral Formula for Derivatives
Useful Results Deducible from the Cauchy Integral Formulas
Evaluation of Improper Integrals by Contour Integration
Taylor and Laurent Series: Residue Theorem and
Applications
Sequences, Series, and Convergence
Uniform Convergence
Power Series
Taylor Series
Laurent Series
Classification of Singularities and Zeros
Residues and the Residue Theorem
Applications of the Residue Theorem
The Laplace Inversion Integral
Conformal Mapping
Geometrical Aspects of Analytic Functions: Mapping
Conformal Mapping
The Linear Fractional Transformation
Mappings by Elementary Functions
The Schwarz-Christoffel Transformation
Boundary Value Problems, Potential Theory, and
Conformal Mapping
Laplace’s Equation and Conformal Mapping – Boundary
Value Problems
Standard Solutions of the Laplace Equation
Steady-State Two-Dimensional Temperature Distribution
Steady Two-Dimensional Fluid Flow
Two-Dimensional Electrostatics
Biography
Alan Jeffrey
“This book is an excellent textbook, well written, and enjoyable. It is warmly recommended to students of applied mathematics and engineering that are interested in various applications of complex analysis.”
— Gabriela Kohr, writing in Zentralblatt MATH, Vol. 1113, 2007
