Complex Variables - Anthony D. Osborne

COMPLEX VARIABLES
AND THEIR APPLICATIONS

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Complex Variables and Their Applications by Anthony D. Osborne has been designed to be versatile, being not only suitable for a first and second course in complex variables for mathematicians, but also with enough applications to be of use and interest to engineering and other science students. It is written specifically with today's undergraduates in mind and only assumes a knowledge of basic real analysis and calculus. The text includes the standard techniques and applications of complex variables, with plentiful examples. The book also presents the important analytical concepts and techniques used in deriving standard results in complex analysis, although readers who are more interested in applications may wish to leave these derivations and go straight to the calculations and examples.

The book can be used at different levels depending on the sections that are chosen. Any material which is normally covered in a first course is included within the first six chapters. The next three chapters deal with applications other than residue theory, including some applications concerning differential equations which are not often given in books on complex variables. Some of the material in Chapters 5, 6, 9, 10, and possibly 11, is suitable for inclusion in a second course. A short bibliography, which suggests further reading, but is by no means exhaustive, is given at the end of the book.

	Contents

1	Functions of a Complex Variable
	Complex Numbers The Complex Plane The Riemann Sphere The Polar Form of a Complex Number Functions of a Complex Variable The Elementary Functions

2	Differentiation and the Cauchy-Riemann Equations
	Limits of Functions Continuity Branch Points and Riemann Surfaces Derivatives Analytic Functions and the Cauchy-Riemann Equations Harmonic Functions Singular Points and Zeros

3	Integration, Cauchy's Theorems and Related Results
	Definite Integrals Cauchy's Theorem Cauchy's Integral Formula Consequences of Cauchy's Integral Formulae The Location of Roots in Equations The Cauchy-Goursat Theorem

4	Infinite Series, Taylor Series and Laurent Series
	Sequences Sequences of Functions Infinite Series Power Series Taylor Series Laurent Series Singular Points

5	The Residue Theorem and its Applications
	Cauchy's Residue Theorem and Calculation of Residues Evaluation of Real Definite Integrals Using the Residue Theorem Evaluation of Other Real Definite Integrals Integrals Involving Branch Points Integrals with an Infinite Number of Singular Points Summation of Series Using the Residue Theorem Partial Fraction Expansions

6	Conformal Transformations
	Conformal Transformations The Existence of Conformal Transformations Bilinear Transformations Cross Ratios Inverse Points Special Elementary Transformations Exponential and Logarithmic Transformations Hyperbolic and Trigonometric Transformations The Schwarz-Christoffel Transformation

7	Linear Ordinary Differential Equations
	Second-Order Linear Equations The Solution of Linear Second-Order Equations in Series Solutions in a Neighbourhood of a Regular Singular Point The Method of Frobenius Equations with Assigned Singularities Special Functions Contour Integral Solutions of Differential Equations

8	Fourier and Laplace Transforms
	Integral Transforms Fourier Transforms and Their Applications Laplace Transforms Applications of Laplace transforms to Ordinary Differential Equations

9	Laplace's Equation and Other Partial Differential Equations
	Harmonic Functions The Dirichlet Problem for the Unit Circle Harmonic Functions and Conformal Transformations The Use of Conformal Mappings in Solving Laplace's Equation Two-Dimensional Fluid Flow The Solution of Linear Partial Differential Equations Using Integral Transforms Separation of Variables

10	Analytic Functions
	Analytic Continuation Analytic Continuation by Means of Taylor Series Analytic Continuation Across a Boundary Infinite Products Weierstrass's Factor Theorem Functions Defined by Integrals The Gamma Function Asymptotic Expansions

11	Elliptic Functions
	Jacobian Elliptic Functions Elliptic Functions in General Introduction to the Weierstrassian Elliptic Function