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Details

Author Theodore W.Gamelin

Press Beijing World Publishing Corporation

Isbn 7506292297

Publication Date May 2008

Pages 478

Size 24k

Edition First

Cover Paperback

Language English

Page material gelatine plate paper

Product Description:

This book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. Students should be familiar with the Cartesian representation of complex numbers and with the algebra of complex numbers, that is, they should know that i2 = -1. A familiarity with multivariable calculus is also required, but here the fundamental ideas are reviewed. In fact, complex analysis provides a good training ground for multivariable calculus. It allows students to consolidate their understanding of parametrized curves, tangent vectors, arc length, gradients, line integrals, independence of path, and Green's theorem. The ideas surrounding independence of path are particularly difficult for students in calculus, and they are not absorbed by most students until they are seen again in other courses.

Content:

Preface

Introduction

First part

Chapter 1 The Complex Plane and Elementary Functions

1. Complex Numbers

2. Polar Representation

3. Stereographic Projection

4. The Square and Square Root Functions

5. The Exponential Function

6. The Logarithm Function

7. Power Functions and Phase Factors

8. Trigonometric and Hyperbolic Functions

Chapter 2 Analytic Functions

1. Review of Basic Analysis

2. Analytic Functions

3. The Cauchy-Riemann Equations

4. Inverse Mappings and the Jacobian

5. Harmonic Functions

6. Conformal Mappings

7. Fractional Linear Transformations

Chapter 3 Line Integrals and Harmonic Functions

1. Line Integrals and Green's Theorem

2. Independence of Path

3. Harmonic Conjugates

4. The Mean Value Property

5. The Maximum Principle

6. Applications to Fluid Dynamics

7. Other Applications to Physics

Chapter 4 Complex Integration and Analyticity

1. Complex Line Integrals

2. Fundamental Theorem of Calculus for Analytic Functions

3. Cauchy's Theorem

4. The Cauchy Integral Formula

5. Liouville's Theorem

6. Morera's Theorem

7. Goursat's Theorem

8. Complex Notation and Pompeiu's Formula

Chapter 5 Power Series

1. Infinite Series

2. Sequences and Series of Functions

3. Power Series

4. Power Series Expansion of an Analytic Function

5. Power Series Expansion at Infinity

6. Manipulation of Power Series

7. The Zeros of an Analytic Function

8. Analytic Continuation

Chapter 6 Laurent Series and Isolated Singularities

1. The Laurent Decomposition

2. Isolated Singularities of an Analytic Function

3. Isolated Singularity at Infinity

4. Partial Fractions Decomposition

5. Periodic Functions

6. Fourier Series

Chapter 7 The Residue Calculus

1. The Residue Theorem

2. Integrals Featuring Rational Functions

3. Integrals of Trigonometric Functions

4. Integrands with Branch Points

5. Fractional Residues

6. Principal Values

7. Jordan's Lemma

8. Exterior Domains

Second part

Chapter 8 The Logarithmic Integral

1. The Argument Principle

2. Rouche's Theorem

3. Hurwitz's Theorem

4. Open Mapping and Inverse Function Theorems

5. Critical Points

6. Winding Numbers

7. The Jump Theorem for Cauchy Integrals

8. Simply Connected Domains

Chapter 9 The Schwarz Lemma and Hyperbolic Geometry

1. The Schwarz Lemma

2. Conformal Self-Maps of the Unit Disk

3. Hyperbolic Geometry

Chapter 10 Harmonic Functions and the Reflection Principle

1. The Poisson Integral Formula

2. Characterization of Harmonic Functions

3. The Schwarz Reflection Principle

Chapter 11 Conformal Mapping

1. Mappings to the Unit Disk and Upper Half-Plane

2. The Riemann Mapping Theorem

3. The Schwarz-Christoffel Formula

4. Return to Fluid Dynamics

5. Compactness of Families of Functions

6. Proof of the Riemann Mapping Theorem

Third part

Chapter 12 Compact Families of Meromorphic Functions

1. Marty's Theorem

2. Theorems of Montel and Picard

3. Julia Sets

4. Connectedness of Julia Sets

5. The Mandelbrot Set

Chapter 13 Approximation Theorems

1. Runge's Theorem

2. The Mittag-Leffler Theorem

3. Infinite Products

4. The Weierstrass Product Theorem

Chapter 14 Some Special Functions

1. The Gamma Function

2. Laplace Transforms

3. The Zeta Function

4. Dirichlet Series

5. The Prime Number Theorem

Chapter 15 The Dirichlet Problem

1. Green's Formulae

2. Subharmonic Functions

3. Compactness of Families of Harmonic Functions

4. The Perron Method

5. The Riemann Mapping Theorem Revisited

6. Green's Function for Domains with Analytic Boundary

7. Green's Function for General Domains

Chapter 16 Riemann Surfaces

1. Abstract Riemann Surfaces

2. Harmonic Functions on a Riemann Surface

3. Green's Function of a Surface

4. Symmetry of Green's Function

5. Bipolar Green's Function

6. The Uniformization Theorem

7. Covering Surfaces

Hints and Solutions for Selected Exercises

References

List of Symbols

Index

……

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