Complex Analysis

Front Cover
Springer, Mar 14, 2013 - Mathematics - 458 pages
The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read ing material for students on their own. A large number of routine exer cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

CHAPTER
3
1 Definition
9
4 Limits and Compact Sets
17
5 Complex Differentiability
27
CHAPTER VIII
31
7 Angles Under Holomorphic Maps
33
2 Convergent Power Series
47
3 Relations Between Formal and Convergent Series
60
5 Fractional Linear Transformations
227
Harmonic Functions
237
2 Examples
247
3 Basic Properties of Harmonic Functions
254
4 The Poisson Formula
264
PART
276
3 Application of Schwarz Reflection
287
2 Compact Sets in Function Spaces
293

4 Analytic Functions
68
6 The Inverse and Open Mapping Theorems
76
7 The Local Maximum Modulus Principle
83
2 Integrals Over Paths
94
3 Local Primitive for a Holomorphic Function
104
4 Another Description of the Integral Along a Path
110
5 The Homotopy Form of Cauchys Theorem
116
7 The Local Cauchy Formula
126
CHAPTER IV
133
3 Artins Proof
149
CHAPTER V
156
3 Isolated Singularities
165
CHAPTER VI
173
2 Evaluation of Definite Integrals
191
CHAPTER VII
208
3 The Upper Half Plane
215
4 Behavior at the Boundary
299
CHAPTER XI
307
2 The Dilogarithm
315
PART THREE
321
2 The PicardBorel Theorem
330
3 Bounds by the Real Part BorelCarathéodory Theorem
338
5 Entire Functions with Rational Values
344
CHAPTER XIII
351
3 Functions of Finite Order
366
3 The Addition Theorem
383
2 The Gamma Function
396
3 The Lerch Formula
412
2 The Main Lemma and its Application
428
3 Analytic Differential Equations
442
Index
455
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information