Complex Made Simple

David C. Ullrich

ISBN 10: 0821844792 / ISBN 13: 9780821844793

Editore: American Mathematical Society, US, 2008

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Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from? In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work. Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains.Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem. This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.

Codice articolo LU-9780821844793

Dati bibliografici

• Titolo: Complex Made Simple
• Editore: American Mathematical Society, US
• Data di pubblicazione: 2008
• Lingua: Inglese
• ISBN 10: 0821844792
• ISBN 13: 9780821844793
• Rilegatura: Hardback
• Condizione: New
• Dimensioni: 19,05 L x 3,18 A x 26,04 P centimetri 19,05 centimetri di larghezza per 3,18 centimetri di altezza per 26,04 centimetri di profondità
• Peso: 1062 grammi

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Riassunto

Ullrich teaches readers how to think like analysts long before they get to the "Big Picard Theorem," explaining why proofs do and do not work. He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Carathéodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems. He continues with Abel's theorem, the gamma function, universal covering spaces, Cauchy's theorem for non-holomorphic functions and harmonic conjugates. The result is suitable for a course in complex analysis for students with experience in advanced calculus. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)

Product Description

Book by David C Ullrich