Complex Analysis / Edition 4

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Now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while 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 is found in other texts, and the resulting proofs often shed more light on the results than the standard proofs. While the first part is suitable for an introductory course at undergraduate level, the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course.

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Editorial Reviews

In this new edition of a standard textbook, new exercises have been added, in addition to more material on the Borel theorem, Picard's theorem, and J.D. Newman's proof of the prime number theorem. The treatment of gamma and zeta functions has been expanded and an appendix has been added which includes material not usually included in standard texts. The first part of the book is an introduction to complex analysis, while the second covers many special topics which may be used in an advanced course. Annotation c. Book News, Inc., Portland, OR (
From the Publisher
"The very understandable style of explanation, which is typical for this author, makes the book valuable for both students and teachers."
EMS Newsletter, Vol. 37, Sept. 2000

Fourth Edition

S. Lang

Complex Analysis

"A highly recommendable book for a two semester course on complex analysis."


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Product Details

  • ISBN-13: 9780387985923
  • Publisher: Springer New York
  • Publication date: 12/7/1998
  • Series: Graduate Texts in Mathematics Series, #103
  • Edition description: 4th ed. 1999. Corr. 3rd printing 2003
  • Edition number: 4
  • Pages: 489
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.06 (d)

Table of Contents

Pt. 1 Basic Theory 1
Ch. I Complex Numbers and Functions 3
Ch. II Power Series 37
Ch. III Cauchy's Theorem, First Part 86
Ch. IV Winding Numbers and Cauchy's Theorem 133
Ch. V Applications of Cauchy's Integral Formula 156
Ch. VI Calculus of Residues 173
Ch. VII Conformal Mappings 208
Ch. VIII Harmonic Functions 241
Pt. 2 Geometric Function Theory 291
Ch. IX Schwarz Reflection 293
Ch. X The Riemann Mapping Theorem 306
Ch. XI Analytic Continuation Along Curves 322
Pt. 3 Various Analytic Topics 337
Ch. XII Applications of the Maximum Modulus Principle and Jensen's Formula 339
Ch. XIII Entire and Meromorphic Functions 372
Ch. XIV Elliptic Functions 391
Ch. XV The Gamma and Zeta Functions 408
Ch. XVI The Prime Number Theorem 440
App. 1 Summation by Parts and Non-Absolute Convergence 453
App. 2 Difference Equations 457
App. 3 Analytic Differential Equations 461
App. 4 Fixed Points of a Fractional Linear Transformation 465
App. 5 Cauchy's Formula for C[superscript infinity] Functions 467
App. 6 Cauchy's Theorem for Locally Integrable Vector Fields 472
Bibliography 479
Index 481
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