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Handbook of complex variables. (English) Zbl 0946.30001
Boston, MA: Birkhäuser. xxiv, 290 p. (1999).
This handbook of complex variables is a comprehensive reference work for scientists, students and engineers who need to know and use the basic concepts in complex analysis of one variable. It is not a book of mathematical theory but a book of mathematical practice. All basic ideas of complex analysis and many typical applications are treated.
The book does not develop theory and proofs, but there are given thorough references for all topics. Sometimes there is no strict logical ordering of topics and some topics or formulas have been repeated. But this makes the book very easy to use in practice. It is also written in a very vivid style and it contains many helpful figures and graphs.
The material is organized in 15 chapters whose contents we now give in key words.
Chapter 1. The complex plane: Complex arithmetic, exponential function and applications, holomorphic functions.
Chapter 2. Complex line integrals: Real and complex line integrals, complex differentiability and conformality, Cauchy’s integral theorem and formula.
Chapter 3. Applications of the Cauchy theory: Liouville’s theorem, fundamental theorem of algebra, sequences of holomorphic functions, power series representation of holomorphic functions, zeros of holomorphic functions.
Chapter 4. Isolated singularities and Laurent series: Classification of isolated singularities, Casorati-Weierstrass theorem, Laurent expansions and examples, calculus of residues, calculation of definite integrals and sums, meromorphic functions.
Chapter 5. The argument principle: Counting zeros and poles, open mapping theorem, Rouché’s theorem, Hurwitz’s theorem, maximum principle, Schwarz’s lemma.
Chapter 6. The geometric theory of holomorphic functions: Definition of conformal mappings, conformal mappings of the unit disc, linear fractional transformations, Riemann mapping theorem, conformal mappings of annuli.
Chapter 7. Harmonic functions: Basic properties, maximum principle and mean value property, Poisson’s integral formula, Schwarz’s reflection principle, Harnack’s principle, Dirichlet problem and subharmonic functions.
Chapter 8. Infinite series and products: Basic concepts, Weierstrass’s product and factorization theorem, Mittag-Leffler’s theorem, normal families, Montel’s theorem.
Chapter 9. Applications of infinite sums and products: Jensen’s formula, Blaschke products, Hadamard’s gap theorem, entire functions of finite order.
Chapter 10. Analytic continuation: Analytic function elements, analytic continuation along a curve, monodromy theorem, idea of a Riemann surface, Picard’s theorems.
Chapter 11. Rational approximation theory: Runge’s theorem, Mergelyan’s theorem.
Chapter 12. Special classes of holomorphic functions: Schlicht functions, Bieberbach conjecture, boundary continuation of conformal mappings, Hardy spaces.
Chapter 13. Special functions: Gamma and beta function, Riemann zeta function, counting functions of number theory, prime number theorem.
Chapter 14. Applications that depend on conformal mapping: Dirichlet problem, steady state heat distribution, electrostatics, incompressible fluid flow, numerical techniques of conformal mapping, Schwarz-Christoffel formula, pictorial catalog of conformal maps.
Chapter 15. Transform theory: Fourier series, Fourier transform, Laplace transform, \(z\)-transform.
A 16-th chapter is devoted to some computer packages for studying complex variables.
The book ends with an extensive glossary of key terms, a complete list of notation, a table of Laplace transforms, a guide to the literature, a reference section and a detailed subject index.

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functions of a complex variable