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Introduction to Complex Analysis

ISBN-10: 047133233X

ISBN-13: 9780471332336

Edition: 2000

Authors: O. Carruth McGehee

List price: $196.00
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Description:

This work aims to offer an accessible introduction to the theory of complex analysis - a fundamental branch of mathematics that has important applications in many areas of the physical sciences.
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Book details

List price: $196.00
Copyright year: 2000
Publisher: John Wiley & Sons, Incorporated
Publication date: 9/15/2000
Binding: Hardcover
Pages: 456
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.628
Language: English

Preface
Symbols and Terms
Preliminaries
Preview
It Takes Two Harmonic Functions
Heat Flow
A Geometric Rule
Electrostatics
Fluid Flow
One Model, Many Applications
Exercises
Sets, Functions, and Visualization
Terminology and Notation for Sets
Terminology and Notation for Functions
Functions from R to R
Functions from R[superscript 2] to R
Functions from R[superscript 2] to R[superscript 2]
Exercises
Structures on R[superscript 2], and Linear Maps from R[superscript 2] to R[superscript 2]
The Real Line and the Plane
Polar Coordinates in the Plane
When Is a Mapping M: R[superscript 2] [right arrow] R[superscript 2] Linear?
Visualizing Nonsingular Linear Mappings
The Determinant of a Two-by-Two Matrix
Pure Magnifications, Rotations, and Conjugation
Conformal Linear Mappings
Exercises
Open Sets, Open Mappings, Connected Sets
Distance, Interior, Boundary, Openness
Continuity in Terms of Open Sets
Open Mappings
Connected Sets
Exercises
A Review of Some Calculus
Integration Theory for Real-Valued Functions
Improper Integrals, Principal Values
Partial Derivatives
Divergence and Curl
Exercises
Harmonic Functions
The Geometry of Laplace's Equation
The Geometry of the Cauchy-Riemann Equations
The Mean Value Property
Changing Variables in a Dirichlet or Neumann Problem
Exercises
Basic Tools
The Complex Plane
The Definition of a Field
Complex Multiplication
Powers and Roots
Conjugation
Quotients of Complex Numbers
When Is a Mapping L : C [right arrow] C Linear?
Complex Equations for Lines and Circles
The Reciprocal Map, and Reflection in the Unit Circle
Reflections in Lines and Circles
Exercises
Visualizing Powers, Exponential, Logarithm, and Sine
Powers of z
Exponential and Logarithms
Sin z
The Cosine and Sine, and the Hyperbolic Cosine and Sine
Exercises
Differentiability
Differentiability at a Point
Differentiability in the Complex Sense: Holomorphy
Finding Derivatives
Picturing the Local Behavior of Holomorphic Mappings
Exercises
Sequences, Compactness, Convergence
Sequences of Complex Numbers
The Limit Superior of a Sequence of Reals
Implications of Compactness
Sequences of Functions
Exercises
Integrals Over Curves, Paths, and Contours
Integrals of Complex-Valued Functions
Curves
Paths
Pathwise Connected Sets
Independence of Path and Morera's Theorem
Goursat's Lemma
The Winding Number
Green's Theorem
Irrotational and Incompressible Fluid Flow
Contours
Exercises
Power Series
Infinite Series
The Geometric Series
An Improved Root Test
Power Series and the Cauchy-Hadamard Theorem
Uniqueness of the Power Series Representation
Integrals That Give Rise to Power Series
Exercises
The Cauchy Theory
Fundamental Properties of Holomorphic Functions
Integral and Series Representations
Eight Ways to Say "Holomorphic"
Determinism
Liouville's Theorem
The Fundamental Theorem of Algebra
Subuniform Convergence Preserves Holomorphy
Exercises
Cauchy's Theorem
Cerny's 1976 Proof
Simply Connected Sets
Subuniform Boundedness, Subuniform Convergence
Isolated Singularities
The Laurent Series Representation on an Annulus
Behavior Near an Isolated Singularity in the Plane
Examples: Classifying Singularities, Finding Residues
Behavior Near a Singularity at Infinity
A Digression: Picard's Great Theorem
Exercises
The Residue Theorem and the Argument Principle
Meromorphic Functions and the Extended Plane
The Residue Theorem
Multiplicity and Valence
Valence for a Rational Function
The Argument Principle: Integrals That Count
Exercises
Mapping Properties
Exercises
The Riemann Sphere
Exercises
The Residue Calculus
Integrals of Trigonometric Functions
Exercises
Estimating Complex Integrals
Exercises
Integrals of Rational Functions Over the Line
Exercises
Integrals Involving the Exponential
Integrals Giving Fourier Transforms
Exercises
Integrals Involving a Logarithm
Exercises
Integration on a Riemann Surface
Mellin Transforms
Exercises
The Inverse Laplace Transform
Exercises
Boundary Value Problems
Examples
Easy Problems
The Conformal Mapping Method
Exercises
The Mobius Maps
Exercises
Electric Fields
A Point Charge in 3-Space
Uniform Charge on One or More Long Wires
Examples with Bounded Potentials
Exercises
Steady Flow of a Perfect Fluid
Exercises
Using the Poisson Integral to Obtain Solutions
The Poisson Integral on a Disk
Solutions on the Disk by the Poisson Integral
Geometry of the Poisson Integral
Harmonic Functions and the Mean Value Property
The Neumann Problem on a Disk
The Poisson Integral on a Half-Plane, and on Other Domains
Exercises
When Is the Solution Unique?
Exercises
The Schwarz Reflection Principle
Schwarz-Christoffel Formulas
Triangles
Rectangles and Other Polygons
Generalized Polygons
Exercises
Lagniappe
Dixon's 1971 Proof of Cauchy's Theorem
Runge's Theorem
Exercises
The Riemann Mapping Theorem
Exercises
The Osgood-Taylor-Caratheodory Theorem
References
Index