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A survey of recent developments both in the classical and modern fields of the theory. Contents include: The complex analytic structure of the space of closed Riemann surfaces; Complex analysis on noncompact Riemann domains; Proof of the Teichmuller-Ahlfors theorem; The conformal mapping of Riemann surfaces; On certain coefficients of univalent functions; Compact analytic surfaces; On differentiable mappings; Deformations of complex analytic manifolds. Originally published in 1960.

Functional analysis --- Analytic functions --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's

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This introduction to complex variable methods begins by carefully defining complex numbers and analytic functions, and proceeds to give accounts of complex integration, Taylor series, singularities, residues and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The emphasis is laid on understanding the use of methods, rather than on rigorous proofs. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. This second edition now contains 350 stimulating exercises of high quality, with solutions given to many of them. Material has been updated and additional proofs on some of the important theorems in complex function theory are now included, e.g. the Weierstrass-Casorati theorem. The book is highly suitable for students wishing to learn the elements of complex analysis in an applied context.

Functions of complex variables --- Analytic functions. --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Series, Taylor's --- Complex variables --- Elliptic functions --- Functions of real variables

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Nine Introductions in Complex Analysis

Complex analysis --- Functions of complex variables. --- Analytic functions. --- Functions of complex variables --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Series, Taylor's --- Complex variables --- Elliptic functions --- Functions of real variables

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Analytic functions. --- Hardy spaces. --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Series, Taylor's --- Analyse fonctionnelle --- Functional analysis. --- Espaces de hardy

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The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs. The book has educated a whole generation of mathematicians with backgrounds in complex analysis and function algebras. It has had a great impact on the early careers of many leading analysts and has been widely adopted as a textbook for graduate courses and learning seminars in both the US and abroad. - From the Citation for the 2003 Leroy P. Steele Prize for Exposition The author has not attempted to produce a compendium. Rather, he has selected a range of topics in a many-faceted theory and, within that range, penetrated to considerable depth...the author has succeeded in bringing out the beauty of a theory which, despite its relatively advanced age---now approaching 80 years---continues to surprise and to delight its practitioners. The author has left his mark on the subject. - Donald Sarason, Mathematical Reviews Garnett's Bounded Analytic Functions is to function theory as Zygmund's Trigonometric Series is to Fourier analysis. Bounded Analytic Functions is widely regarded as a classic textbook used around the world to educate today's practioners in the field, and is the primary source for the experts. It is beautifully written, but intentionally cannot be read as a novel. Rather it gives just the right level of detail so that the motivated student develops the requisite skills of the trade in the process of discovering the beauty of the combination of real, complex and functional analysis. - Donald E. Marshall, University of Washington.

Analytic functions. --- Functional analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Functions of complex variables. --- Functions of a Complex Variable. --- Functional Analysis. --- Complex variables --- Elliptic functions --- Functions of real variables