This is the fourth edition of Serge Lang´s Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course. This is a revised edition, new examples and exercises have been added, and many minor improvements have been made throughout the text.
The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The book consists of three parts. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics such as the argument principle, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics selected include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challenging, in all chapters. The book is based on lectures given over the years by the author at several places, particularly the Interuniversity Summer School at Perugia (Italy), and also UCLA, Brown University, Valencia (Spain), and La Plata (Argentina). A native of Minnesota, the author did his undergraduate work at Yale University and his graduate work at UC Berkeley. After spending some time at MIT and at the Universidad Nacional de La Plata (Argentina), he joined the faculty at UCLA in 1968.
In this textbook, a concise approach to complex analysis of one and several variables is presented. After an introduction of Cauchy´s integral theorem general versions of Runge´s approximation theorem and Mittag-Leffler´s theorem are discussed. The fi rst part ends with an analytic characterization of simply connected domains. The second part is concerned with functional analytic methods: Fréchet and Hilbert spaces of holomorphic functions, the Bergman kernel, and unbounded operators on Hilbert spaces to tackle the theory of several variables, in particular the inhomogeneous Cauchy-Riemann equations and the d-bar Neumann operator. ContentsComplex numbers and functionsCauchy´s Theorem and Cauchy´s formulaAnalytic continuationConstruction and approximation of holomorphic functionsHarmonic functionsSeveral complex variablesBergman spacesThe canonical solution operator to Nuclear Fréchet spaces of holomorphic functionsThe -complexThe twisted -complex and Schrödinger operators
A brand new appendix by Oscar Garcia-Prada graces this third edition of a classic work. In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Wells´s superb analysis also gives details of the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths´s period mapping, quadratic transformations, and Kodaira´s vanishing and embedding theorems. Oscar Garcia-Prada´s appendix gives an overview of the developments in the field during the decades since the book appeared.
This book offers an essential textbook on complex analysis. After introducing the theory of complex analysis, it places special emphasis on the importance of Poincare theorem and Hartog´s theorem in the function theory of several complex variables. Further, it lays the groundwork for future study in analysis, linear algebra, numerical analysis, geometry, number theory, physics (including hydrodynamics and thermodynamics), and electrical engineering. To benefit most from the book, students should have some prior knowledge of complex numbers. However, the essential prerequisites are quite minimal, and include basic calculus with some knowledge of partial derivatives, definite integrals, and topics in advanced calculus such as Leibniz´s rule for differentiating under the integral sign and to some extent analysis of infinite series. The book offers a valuable asset for undergraduate and graduate students of mathematics and engineering, as well as students with no background in topological properties.
This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications. The text is adapted to enable maximum flexibility to instructors and to students who may also choose to progress through the material outside of coursework. Detailed examples may be covered in one course, giving the instructor the option to choose those that are best suited for discussion. Examples showcase a variety of problems with completely worked out solutions, assisting students in working through the exercises. The numerous exercises vary in difficulty from simple applications of formulas to more advanced project-type problems. Detailed hints accompany the more challenging problems. Multi-part exercises may be assigned to individual students, to groups as projects, or serve as further illustrations for the instructor. Widely used graphics clarify both concrete and abstract concepts, helping students visualize the proofs of many results. Freely accessible solutions to every-other-odd exercise are posted to the book´s Springer website. Additional solutions for instructors´ use may be obtained by contacting the authors directly.
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level.
This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations - so-called phase portraits - which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions - even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.
Fundamentals of analytic function theory - plus lucid exposition of 5 important applications: potential theory, ordinary differential equations, Fourier transforms, Laplace transforms, and asymptotic expansions. Includes 66 figures.