Complex variables texts vary from purely theoretical to very applied. Although entitled Applied Complex Variables for Scientists and Engineers, this text puts the mathematics first while including applications in every chapter. It is an excellent text for those hoping to understand the mathematics well and how it is used in applications. Concepts are carefully explained and there are many helpful worked examples. Each chapter has numerous exercises, 340 in all, many of which have answers included.
The first chapter introducting complex numbers includes applictions to electrical circuits. The last section of the chapter on Analytic Functions, on harmonic functions, covers steady state temperature distributions and Poisson’s equation. The logarithmic functions section of the next chapter treats temperature distribution in the upper-half plane. After proving the Cauchy-Goursat theorem, the Cauchy integral formulas and other results Chapter 4 on complex integration concludes with a section on potential functions of conservative fields. An application for Laurent series obtains the potential flow over a perturbed circle.
Singularities and Calculus of Residues includes a section of Fourier transforms as well as one on hydrodynamics in potential fluid flows. The final two chapters are Boundary Value Problems and Initial-Boundary Value Problems and Conformal Mapping and Applications. Overall this text combines a clear, thorough treatment of the standard complex variable theory with illustrative applications. It could be used in a variety of courses but those with primarily an engineering or science background should be well-motivated to encounter carefully presented mathematics first.
Art Gittleman (firstname.lastname@example.org) is Professor of Computer Science at California State University Long Beach.
Preface; 1. Complex numbers; 2. Analytic functions; 3. Exponential, logarithmic and trigonometric functions; 4. Complex integration; 5. Taylor and Laurent series; 6. Singularities and calculus of residues; 7. Boundary value problems and initial-boundary value problems; 8. Conformal mappings and applications; Answers to problems; Index.