Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero

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
J 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