Complex Variables and Applications by Brown and Churchill

[micromass]

• Author: James Brown, Ruel Churchill
• Title: Complex Variables and Applications
• Amazon link https://www.amazon.com/dp/0073051942/?tag=pfamazon01-20

Table of Contents:

[LIST]
[*] Preface 
[*] Complex Numbers
[LIST]
[*] Sums and Products
[*] Basic Algebraic Properties
[*] Further Properties
[*] Vectors and Moduli
[*] Complex Conjugates
[*] Exponential Form
[*] Products and Powers in Exponential Form
[*] Arguments of Products and Quotients
[*] Roots of Complex Numbers
[*] Examples
[*] Regions in the Complex Plane
[/LIST]
[*] Analytic Functions 
[LIST]
[*] Functions of a Complex Variable
[*] Mappings
[*] Mappings by the Exponential Function
[*] Limits
[*] Theorems on Limits
[*] Limits Involving the Point at Infinity
[*] Continuity
[*] Derivatives
[*] Differentiation Formulas
[*] Cauchy-Riemann Equations
[*] Sufficient Conditions for Differentiability 
[*] Polar Coordinates
[*] Analytic Functions
[*] Examples
[*] Harmonic Functions
[*] Uniquely Determined Analytic Functions
[*] Reflection Principle
[/LIST]
[*] Elementary Functions
[LIST]
[*] The Exponential Function
[*] The Logarithmic Function
[*] Branches and Derivatives of Logarithms
[*] Some Identities Involving Logarithms
[*] Complex Exponents
[*] Trigonometric Functions
[*] Hyperbolic Functions
[*] Inverse Trigonometric and Hyperbolic Functions
[/LIST]
[*] Integrals
[LIST]
[*] Derivatives of Functions w(t)
[*] Definite Integrals of Functions w(t)
[*] Contours
[*] Contour Integrals
[*] Some Examples
[*] Examples with Branch Cuts
[*] Upper Bounds for Moduli of Contour Integrals 
[*] Antiderivatives
[*] Proof of the Theorem
[*] Cauchy-Goursat Theorem
[*] Proof of the Theorem
[*] Simply Connected Domains
[*] Multiply Connected Domains
[*] Cauchy Integral Formula
[*] An Extension of the Cauchy Integral Formula
[*] Some Consequences of the Extension
[*] Liouville's Theorem and the Fundamental Theorem of Algebra
[*] Maximum Modulus Principle
[/LIST]
[*] Series
[LIST]
[*] Convergence of Sequences
[*] Convergence of Series
[*] Taylor Series
[*] Proof of Taylor's Theorem
[*] Examples
[*] Laurent Series
[*] Proof of Laurent's Theorem
[*] Examples
[*] Absolute and Uniform Convergence of Power Series
[*] Continuity of Sums of Power Series
[*] Integration and Differentiation of Power Series
[*] Uniqueness of Series Representations
[*] Multiplication and Division of Power Series
[/LIST]
[*] Residues and Poles
[LIST]
[*] Isolated Singular Points
[*] Residues
[*] Cauchy's Residue Theorem
[*] Residue at Infinity
[*] The Three Types of Isolated Singular Points
[*] Residues at Poles
[*] Examples
[*] Zeros of Analytic Functions
[*] Zeros and Poles
[*] Behavior of Functions Near Isolated Singular Points
[/LIST]
[*] Applications of Residues
[LIST]
[*] Evaluation of Improper Integrals
[*] Example
[*] Improper Integrals from Fourier Analysis
[*] Jordan's Lemma
[*] Indented Paths
[*] An Indentation Around a Branch Point
[*] Integration Along a Branch Cut
[*] Definite Integrals Involving Sines and Cosines
[*] Argument Principle
[*] Rouché's Theorem
[*] Inverse Laplace Transforms
[*] Examples
[/LIST]
[*] Mapping by Elementary Functions
[LIST]
[*] Linear Transformations
[*] The Transformation w = 1/z
[*] Mappings by 1/z
[*] Linear Fractional Transformations
[*] An Implicit Form
[*] Mappings of the Upper Half Plane
[*] The Transformation w = sin z 
[*] Mappings by z^2 and Branches of z^{1/2}
[*] Square Roots of Polynomials
[*] Riemann Surfaces
[*] Surfaces for Related Functions
[/LIST]
[*] Conformal Mapping
[LIST]
[*] Preservation of Angles
[*] Scale Factors
[*] Local Inverses
[*] Harmonic Conjugates
[*] Transformations of Harmonic Functions
[*] Transformations of Boundary Conditions
[/LIST]
[*] Applications of Conformal Mapping
[LIST]
[*] Steady Temperatures
[*] Steady Temperatures in a Half Plane
[*] A Related Problem
[*] Temperatures in a Quadrant
[*] Electrostatic Potential
[*] Potential in a Cylindrical Space
[*] Two-Dimensional Fluid Flow
[*] The Stream Function
[*] Flows Around a Corner and Around a Cylinder
[/LIST]
[*] The Schwarz--Christoffel Transformation
[LIST]
[*] Mapping the Real Axis Onto a Polygon
[*] Schwarz--Christoffel Transformation
[*] Triangles and Rectangles
[*] Degenerate Polygons
[*] Fluid Flow in a Channel Through a Slit
[*] Flow in a Channel With an Offset
[*] Electrostatic Potential About an Edge of a Conducting Plate
[/LIST]
[*] Integral Formulas of the Poisson Type
[LIST]
[*] Poisson Integral Formula
[*] Dirichlet Problem for a Disk
[*] Related Boundary Value Problems
[*] Schwarz Integral Formula
[*] Dirichlet Problem for a Half Plane
[*] Neumann Problems
[/LIST]
[*] Appendixes
[LIST]
[*] Bibliography
[*] Table of Transformations of Regions
[/LIST]
[*] Index 
[/LIST]

 

[fourier jr]

Solid user-friendly intro book on complex variables, sort of a rough equivalent to linear algebra done right.

 

[Jorriss]

I'd highly recommend for a physics or chemistry student who wants a better grasp of residues and conformal mappings, but which is not too mathematically intense.

 

[Sankaku]

This is a decent introductory book, geared well toward physicists and engineers but rigorous enough to not offend a math-major. It is clearly written and organized, making it decent as a reference book.

However, it doesn't give much insight into the beauty of the subject. This is a common required textbook for intro courses, so I recommend also getting Visual Complex Analysis as an interesting supplementary text.

There is also a solid free textbook here:
http://www.math.uiuc.edu/~r-ash/CV.html

I am familliar with the author's abstract algebra book (which I like), but I have only skimmed parts of this one.

 

[jasonRF]

I am familiar with the 5th edition, which I have had for about 20 years. I think it is a reasonable book that covers most of the material many engineers/physicists need for routine applications. By far my favorite sections are those on conformal mapping - the presentation is quite good for applications. In most ways Churchill and Brown is better than the book I had to buy for my complex analysis class (introduction to complex analysis, by Priestley), but I am not wild about the organization of most of the book. Overall I prefer the book by Saff and Snider for an introduction.

 

[Petek]

I studied the first ten chapters of this book (omitting the final chapters on applications). I used the seventh edition (the eighth edition is now the most current one). I found this text to be ideal for someone (such as myself) who had studied complex analysis a long time ago and wanted to relearn it. The text is almost entirely free of typos and errors, which I find to be important for self-study. The proofs are clear and the examples well-chosen. The exercises are primarily calculations, with a few simple proofs thrown in. This is not a criticism; I found some of the calculations to be challenging. Someone who wants to study more advanced topics in Complex Analysis should consult a second book, such as Lang, Ahlfors or Conway (after studying this text).