This treatment of complex analysis focuses on function theory on a finitely connected planar domain. It emphasizes domains bounded by a finite number of disjoint analytic simple closed curves. 1983 edition.

Numerous examples and exercises highlight this unified treatment of the Hermitian operator theory in its Hilbert space setting. Its simple explanations of difficult subjects make it intuitively appealing to students in applied mathematics, physics, and engineering. It is also a fine reference for professionals. 1990 edition.

This is the proceedings of the meeting entitled "The 12th MSJ International Research Institute of the Mathematical Society of Japan 2003". The papers cover several important topics in Singularity theory. Especially some of them are survey on motivic integrations, Thom polynomials, complex analytic singularity theory, generic differential geometry etc.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

Complex analysis, more than almost any other undergraduate topic in mathematics, runs the full pure/applied gamut from the most subtle, difficult, and ingenious proofs to the most direct, hands-on, engineering-based applications. This creates challenges for the instructor as much as for the very wide range of students whose various programmes require a secure grasp of complex analysis. Its techniques are indispensable to many, but skill in the use of a mathematical tool is hazardous and fallible without a sound understanding of why and when that tool is the right one to pick up. This kind of understanding develops only by combining careful exploration of ideas, analysis of proofs, and practice across a range of exercises. Integration with Complex Numbers: A Primer on Complex Analysis offers a reader-friendly contemporary balance between idea, proof, and practice, informed by several decades of classroom experience and a seasoned understanding of the backgrounds, motivation, and competing time pressures of today's student cohorts. To achieve its aim of supporting and sustaining such cohorts through those aspects of complex analysis that they encounter in first and second-year study, it also balances competing needs to be self-contained, comprehensive, accessible, and engaging - all in sufficient but not in excessive measures. In particular, it begins where most students are likely to be, and invests the time and effort that are required in order to deliver accessibility and introductory gradualness.

Classic Complex Analysis is a text that has been developed over decades of teaching with an enthusiastic student reception. The first half of the book focuses on the core material. An early chapter on power series gives the reader concrete examples of analytic functions and a review of calculus. Mobius transformations are presented with emphasis on the geometric aspect, and the Cauchy theorem is covered in the classical manner. The remaining chapters provide an elegant and solid overview of special topics such as Entire and Meromorphic Functions, Analytic Continuation, Normal Families, Conformal Mapping, and Harmonic Functions.

Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic. Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension. Topics include power series in one variable, holomorphic functions, Cauchy's integral, more. Exercises. 1973 edition.

Discrete Fourier Analysis.

Cauchy Integrals.

Potential Theory in the Plane.

Construction of Conformal Maps: Simply Connected Regions.

Construction of Conformal Maps for Multiply Connected Regions.

Polynomial Expansions and Conformal Maps.

Univalent Functions.

Bibliography.

Index.

Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions.

Dieses Arbeitsbuch enthalt die Aufgaben, Hinweise, Loesungen und Loesungswege zu allen sechs Teilen des Lehrbuchs Arens et al., Mathematik. Die Inhalte des Buchs stehen als PDF-Dateien auf der Website des Verlags zur Verfugung. Durch die stufenweise Offenlegung der Loesungen ist das Werk bestens geeignet zum Selbststudium, zur Vorlesungsbegleitung und als Prufungsvorbereitung. Inhaltlich spannt sich der Bogen von elementaren Grundlagen uber die Analysis einer Veranderlichen, der linearen Algebra, der Analysis mehrerer Veranderlicher bis hin zu fortgeschrittenen Themen der Analysis, die fur die Anwendung besonders wichtig sind, wie partielle Differenzialgleichungen, Fourierreihen und Laplacetransformationen. Auch eine Vielzahl von Aufgaben zur Wahrscheinlichkeitsrechnung und Statistik ist enthalten.

The "Wiley Classics Library" consists of selected books originally published by John Wiley & Sons that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hope to extend the life of these important works by making them available to future generations of mathematicians and scientists.

Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions.

Develops the higher parts of function theory in a unified presentation. Starts with elliptic integrals and functions and uniformization theory, continues with automorphic functions and the theory of abelian integrals and ends with the theory of abelian functions and modular functions in several variables. The last topic originates with the author and appears here for the first time in book form.

Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries.

This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications. As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four. This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.

Ziel dieses Lehrbuches ist es, einen verstandlichen, moeglichst direkten und in sich geschlossenen Zugang zu wichtigen Ergebnissen der mehrdimensionalen Funktionentheorie zu geben. Hierbei fuhrt der Weg von elementaren Eigenschaften holomorpher Funktionen uber analytische Mengen und Holomorphiebereiche bis hin zum Levi-Problem. Ein abschliessendes Kapitel enthalt mit der Konstruktion des mehrdimensionalen holomorphen Funktionalkalkuls nach Shilov, Waelbroeck und Arens-Calderon und dem Satz von Arens-Royden wichtige Anwendungen auf die Theorie komplexer Banachalgebren. Zahlreiche UEbungsaufgaben erganzen den theoretischen Teil. Vorausgesetzt wird nur der Inhalt der Grundvorlesungen in Analysis und einer ublichen einsemestrigen Vorlesung uber Funktionentheorie einer komplexen Veranderlichen. Das Buch richtet sich besonders an fortgeschrittene Bachelorstudierende oder Studierende eines Masterstudienganges und eignet sich bestens als Begleitlekture zu einer Vorlesung oder auch zum Selbststudium.

'I very much enjoyed reading this book ... Each chapter comes with well thought-out exercises, solutions to which are given at the end of the chapter. Conformal Maps and Geometry presents key topics in geometric function theory and the theory of univalent functions, and also prepares the reader to progress to study the SLE. It succeeds admirably on both counts.'MathSciNetGeometric function theory is one of the most interesting parts of complex analysis, an area that has become increasingly relevant as a key feature in the theory of Schramm-Loewner evolution.Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry.It offers a unique view of the field, as it is one of the first to discuss general theory of univalent maps at a graduate level, while introducing more complex theories of conformal invariants and extremal lengths. Conformal Maps and Geometry is an ideal resource for graduate courses in Complex Analysis or as an analytic prerequisite to study the theory of Schramm-Loewner evolution.

Das Buch bietet eine vollstandige Darstellung der Funktionentheorie, beginnend mit der Theorie der Riemann`schen Flachen einschliesslich Uniformisierungstheorie sowie einer ausfuhrlichen Darstellung der Theorie der kompakten Riemann`schen Flachen, Riemann-Roch`schem Satz, Abel`schem Theorem und Jacobi`schem Umkehrtheorem. Hierdurch motiviert wird eine kurze Einfuhrung in die Funktionentheorie mehrerer Veranderlicher gegeben und dann die Theorie der Abel`schen Funktionen bis hin zum Thetasatz entwickelt. Daran anschliessend und hierdurch motiviert wird eine Einfuhrung in die Theorie der hoeheren Modulfunktionen gegeben.

Questo testo, giunto alla seconda edizione, e adatto per una prima esposizione della teoria delle funzioni di singola variabile complessa e si rivolge a studenti di Fisica, Matematica e Ingegneria che abbiano acquisito le nozioni fondamentali dell Analisi Matematica reale. L esigenza di una nuova pubblicazione nasce dall idea di effettuare una selezione di argomenti, ritenuti fondamentali, la cui esposizione risulti sistematica e autoconsistente in circa 60 ore di lezione mantenendo, al tempo stesso, il rigore matematico volto a favorire la maturazione scientifica dello studente e prepararlo alla lettura di testi avanzati. A corredo della trattazione teorica vengono proposti circa 250 esercizi, raccolti tra le prove scritte assegnate per il superamento del corso, tutti forniti di soluzione dettagliata. Il loro svolgimento costituisce una parte imprescindibile per l acquisizione della materia."

Die Funktionentheorie einer komplexen Variablen hat heute hAher-dimensionale Analoga: dabei wird die Algebra der komplexen Zahlen durch die nicht-kommutative Algebra der reellen Quaternionen bzw. Clifford-Algebren ersetzt. In den letzten 30 Jahren hat sich die so genannte Quaternionen- und die reelle Clifford-Analysis erfolgreich entwickelt. Eine Vielzahl von Anwendungen haben diese Funktionentheorie hAher-dimensionaler Variablen zu einem wichtigen Instrument der Analysis und deren Anwendungen in der mathematischen Physik werden lassen.

Das Buch reflektiert den neuesten Stand der Forschung und entwickelt sowohl die hAher-dimensionalen Ergebnisse als auch die klassischen komplexen Resultate aus einem einheitlichen Begriff der Holomorphie. Der fundamentale Begriff der holomorphen Funktion als LAsung des Cauchy-Riemann-Systems wird im HAher-dimensionalen unter Beibehaltung der Bezeichnung als LAsung eines entsprechenden Systems partieller Differentialgleichungen 1. Ordnung verstanden.

Historische Bemerkungen, zahlreiche Beispiele, viele Abbildungen sowie eine angemessene Auswahl von Aoebungsaufgaben festigen und erweitern die erworbenen Kenntnisse.

Das vorliegende Buch ist fA1/4r Studenten der Mathematik, Physik und mathematisch orientierten Ingenieurstudenten im Grund- und Fachstudium geeignet. Es kann auch als Grundlage von Proseminaren oder Seminaren dienen.

Die beiliegende CD enthAlt eine umfangreiche Literaturdatenbank sowie ein Maple-Package, das die im Buch eingefA1/4hrten Werkzeuge und Methoden als Kommandos bzw. vorgefertigte Prozeduren enthAlt. Einige Beispiel-Worksheets unterstA1/4tzen die Einarbeitung in das Package.

Die ersten vier Kapitel dieser Darstellung der klassischen Funktionentheorie vermitteln mit minimalem Begriffsaufwand und auf geringen Vorkenntnissen aufbauend zentrale Ergebnisse und Methoden der komplexen Analysis einer Veranderlichen und gipfeln in einem Beweis des kleinen Riemannschen Abbildungssatzes und einer Charakterisierung einfach zusammenhangender Gebiete. Weiter werden behandelt: Elliptische Funktionen (Weierstrassscher und Jacobischer Ansatz), die elementare Theorie der Modulformen einer Variablen, Anwendungen der Funktionentheorie auf die Zahlentheorie (einschliesslich eines Beweises des Primzahlsatzes).

Die optisch ubersichtliche Aufbereitung und uber vierhundert Ubungsaufgaben von unterschiedlichstem Schwierigkeitsgrad mit Losungshinweisen machen den Band auch zur Prufungsvorbereitung und zum Selbststudium fur Mathematiker und Physiker gut geeignet.

In der vorliegende vierten Auflage wurden u.a. einige Textstellen uberarbeitet und neue Ubungsaufgaben aufgenommen."

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.

Das neue Lehrbuch zur Funktionentheorie, im Aufbau angelehnt an das klassische Werk von Lars Ahlfors, ist als begleitender Text fur eine einsemestrige einfuhrende Vorlesung konzipiert. Daruber hinaus behandelt der Autor aber auch einige weiterfuhrende Themen. Der Band enthalt zahlreiche Ubungen sowie im Anhang drei Kapitel uber den Zusammenhangsbegriff, kompakte metrische Raume und harmonische Funktionen. Besonderen Wert legt der Autor darauf, alle Beweise elegant und vollstandig darzustellen. Der Band eignet sich auch fur das Selbststudium."