An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician. The author includes numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. In addition to standard topics, readers will find Eisenstein's proof of Euler's product formula for the sine function; Wielandts uniqueness theorem for the gamma function; Stirlings formula; Isssas theorem; Besses proof that all domains in C are domains of holomorphy; Wedderburns lemma and the ideal theory of rings of holomorphic functions; Estermanns proofs of the overconvergence theorem and Blochs theorem; a holomorphic imbedding of the unit disc in C3; and Gausss expert opinion on Riemanns dissertation. Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, combine to make an invaluable source for students and teachers alike
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This book is an ideal text for an advanced course in the theory of complex functions. The author successfully enables the reader to experience function theory personally and to participate in the work of the creative mathematician. Unlike the first volume, it contains numerous glimpses of the function theory of several complex variables emphasizing how autonomous this discipline has become. Covered are Weierstrass' product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's theorems on approximations of analytic functions.
Remmert elegantly breaks the material down into small intelligible sections, with perfectly compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, historical remarks (e.g. Gauss' review of Riemann's thesis), and an extensive list of references will make this book an invaluable source. -- Book Description
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Text: English (translation)
Original Language: German
Original Language: German
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First Sentence:
Infinite products first appeared in 1579 in the work of F. Vieta (Opera, p. 400, Leyden, 1646); he gave the formula . . . for (cf. [Z], p. 104 and p. 118). Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal part distribution, functions with prescribed principal parts, visible boundary points, finite holomorphic maps, winding map, partial fraction series, admissible expansion, null homologous, open compact sets, injection theorem, pentagonal number theorem, square root property, lacunary series, sine product, normal convergence, gap theorem, step polygon, compact convergence, principal ideal theorem, product theorem, inner maps, separation lemma, duplication formula, holornorphic functions, prescribed zeros Key Phrases - Capitalized Phrases (CAPs): (learn more)
Acta Math, New York, Boundary Behavior of Power Series, The Theorems of Bloch, Chelsea Publ, Discussion of the Product Theorem, Euler Partition Products, London Math, Jacobi's Product Representation of the Series, Julius Springer, Petersburg Academy, The Koebe Families, Theory of Stein Spaces New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Infinite products first appeared in 1579 in the work of F. Vieta (Opera, p. 400, Leyden, 1646); he gave the formula . . . for (cf. [Z], p. 104 and p. 118). Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal part distribution, functions with prescribed principal parts, visible boundary points, finite holomorphic maps, winding map, partial fraction series, admissible expansion, null homologous, open compact sets, injection theorem, pentagonal number theorem, square root property, lacunary series, sine product, normal convergence, gap theorem, step polygon, compact convergence, principal ideal theorem, product theorem, inner maps, separation lemma, duplication formula, holornorphic functions, prescribed zeros Key Phrases - Capitalized Phrases (CAPs): (learn more)
Acta Math, New York, Boundary Behavior of Power Series, The Theorems of Bloch, Chelsea Publ, Discussion of the Product Theorem, Euler Partition Products, London Math, Jacobi's Product Representation of the Series, Julius Springer, Petersburg Academy, The Koebe Families, Theory of Stein Spaces New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
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97
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- Theory of Stein Spaces (Classics in Mathematics) by Hans Grauert on page 105, page 124, and page 142
- Analytic Functions by Stanislaw Saks on page 165, page 288, and page 319
- Cyclotomic Fields (Graduate Texts in Mathematics) by S. Lang in Front Matter, Back Matter (1), and Back Matter (2)
- Complex Analysis in One Variable by Raghavan Narasimhan on page 143, page 202, and page 307
- Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis (Encyclopaedia of Mathematical Sciences) by H. Grauert in Front Matter, and page 123
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- Galois Theory (Graduate Texts in Mathematics) by Harold M. Edwards in Back Matter
- Geometric Function Theory in One and Higher Dimensions (Pure and Applied Mathematics, Vol 255) (Chapman & Hall/CRC Pure and Applied Mathematics) by Ian Graham in Back Matter
- Algebraic Geometry (Graduate Texts in Mathematics) by Robin Hartshorne in Front Matter
- An Introduction to Number Theory (Graduate Texts in Mathematics) by G. Everest in Back Matter
- Functions of One Complex Variable (Graduate Texts in Mathematics - Vol 11) (v. 1) by John B Conway in Back Matter
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