Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions.
Author(s): Peter Henrici
Publisher: John Wiley & Sons Inc; American First edition
Title Page
Preface
Contents
1 Formal Power Series
1.1 Algebraic Preliminaries: Complex Numbers
1.2 Definition and Algebraic Properties of Formal Power Series
1.3 A Matrix Representation of Formal Power Series
1.4 Differentiation of Formal Power Series
1.5 Formal Hypergeometric Series and Finite Hypergeometric Sums
1.6 The Composition of a Formal Power Series with a Nonunit
1.7 The Group of Almost Units Under Composition
1.8 Formal Laurent Series: Residues
1.9 The Lagrange-Burmann Theorem
Seminar Assignments
Notes
2 Functions Analytic at a Point
2.1 Banach Algebras: Functions
2.2 Convergent Power Series
2.3 Functions Analytic at a Point
2.4 Composition and Inversion of Analytic Functions
2.5 Elementary Transcendental Functions
2.6 Matrix-Valued Functions
2.7 Sequences of Functions Analytic at a Point
Seminar Assignments
Notes
3 Analytic Continuation
3.1 Rearrangement of Power Series; Derivatives
3.2 Analytic Extension and Continuation
3.3 New Determination of the Radius of Convergence
3.4 Sequences of Functions Analytic in a Region
3.5 Analytic Continuation Along an Arc: Monodromy Theorem
3.6 Numerical Analytic Continuation Along an Arc
Seminar Assignments
Notes
4 Complex Integration
4.1 Complex Functions of a Real Variable
4.2 The Integral of a Function Along an Arc
4.3 Integrals of Analytic Functions
4.4 The Laurent Series: Isolated Singularities
4.5 Applications of the Laurent Series: Bessel Functions, Fourier Series
4.6 Continuous Argument, Winding Number, Jordan Curve Theorem
4.7 Residue Theorem: Cauchy Integral Formula
4.8 Applications of the Residue Theorem: Evaluation of Definite Integrals
4.9 Applications of the Residue Theorem: Summation of Infinite Series
4.10 The Principle of the Argument
Seminar Assignments
Notes
5 Conformal Mapping
5.1 Geometric Interpretation of Complex Functions
5.2 Moebius Transformations: Algebraic Theory
5.3 Moebius Transformations: The Riemann Sphere
5.4 Moebius Transformations: Symmetry
5.5 Holomorphic Functions and Conformal Maps
5.6 Conformal Transplants
5.7 Problems of Plane Electrostatics
5.8 Two-Dimensional Ideal Flows
5.9 Poisson's Equation
5.10 General Results on Conformal Maps
5.11 Symmetry
5.12 The Schwarz-Christoffel Mapping Function
5.13 Mapping the Rectangle. An Elliptic Integral
5.14 Rounding Corners in Schwarz-Christoffel Maps
Seminar Assignment
Notes
6 Polynomials
6.1 The Horner Algorithm
6.2 Sign Changes. The Rule of Descartes
6.3 Cauchy Indices: The Number of Zeros of a Real Polynomial in a Real Interval
6.4 Disks Containing a Specified Number of Zeros
6.5 Geometry of Zeros (Theorems of Gauss-Lucas, Laguerre, and Grace)
6.6 Circular Arithmetic
6.7 The Number of Zeros in a Half-Plane
6.8 The Number of Zeros in a Disk
6.9 Methods for Determining Zeros: A Survey
6.10 Methods of Search for a Single Zero
6.11 Methods of Search and Exclusion for the Simultaneous Determination of All Zeros
6.12 Fixed Points of Analytic Functions: Iteration
6.13 Newton's Method for Polynomials
6.14 Methods of Descent
Seminar Assignments
Notes
7 Partial Fractions
7.1 Construction of the Partial Fraction Representation of a Rational Function
7.2 Partial Fractions: Miscellaneous Applications
7.3 Some Applications to Combinatorial Analysis
7.4 Difference Equations
7.5 Hankel Determinants
7.6 The Quotient-Difference Algorithm
7.7 Hadamard Polynomials
7.8 Matrix Interpretation
7.9 Poles with Equal Moduli
7.10 Partial Fraction Expansions of Meromorphic Functions
Seminar Assignments
Notes
Bibliography
Index
