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Description:

Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to make students comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

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Book details

List price: $99.99 Copyright year: 2007 Publisher: Springer Berlin / Heidelberg Publication date: 11/10/2006 Binding: Hardcover Pages: 319 Size: 6.10" wide x 9.25" long x 0.37" tall Weight: 1.452 Language: English

AuthorTable of Contents

Complex Analysis

Complex Numbers

Our Number System

Addition and Multiplication of Integers

Inverse Operations

Negative Numbers

Fractional Numbers

Irrational Numbers

Imaginary Numbers

Logarithm

Napier's Idea of Logarithm

Briggs' Common Logarithm

A Peculiar Number Called e

The Unique Property of e

The Natural Logarithm

Approximate Value of e

The Exponential Function as an Infinite Series

Compound Interest

The Limiting Process Representing e

The Exponential Function e[superscript x]

Unification of Algebra and Geometry

The Remarkable Euler Formula

The Complex Plane

Polar Form of Complex Numbers

Powers and Roots of Complex Numbers

Trigonometry and Complex Numbers

Geometry and Complex Numbers

Elementary Functions of Complex Variable

Exponential and Trigonometric Functions of z

Hyperbolic Functions of z

Logarithm and General Power of z

Inverse Trigonometric and Hyperbolic Functions

Exercises

Complex Functions

Analytic Functions

Complex Function as Mapping Operation

Differentiation of a Complex Function

Cauchy-Riemann Conditions

Cauchy-Riemann Equations in Polar Coordinates

Analytic Function as a Function of z Alone

Analytic Function and Laplace's Equation

Complex Integration

Line Integral of a Complex Function

Parametric Form of Complex Line Integral

Cauchy's Integral Theorem

Green's Lemma

Cauchy-Goursat Theorem

Fundamental Theorem of Calculus

Consequences of Cauchy's Theorem

Principle of Deformation of Contours

The Cauchy Integral Formula

Derivatives of Analytic Function

Exercises

Complex Series and Theory of Residues

A Basic Geometric Series

Taylor Series

The Complex Taylor Series

Convergence of Taylor Series

Analytic Continuation

Uniqueness of Taylor Series

Laurent Series

Uniqueness of Laurent Series

Theory of Residues

Zeros and Poles

Definition of the Residue

Methods of Finding Residues

Cauchy's Residue Theorem

Second Residue Theorem

Evaluation of Real Integrals with Residues

Integrals of Trigonometric Functions

Improper Integrals I: Closing the Contour with a Semicircle at Infinity

Fourier Integral and Jordan's Lemma

Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour

Integration Along a Branch Cut

Principal Value and Indented Path Integrals

Exercises

Determinants and Matrices

Determinants

Systems of Linear Equations

Solution of Two Linear Equations

Properties of Second-Order Determinants

Solution of Three Linear Equations

General Definition of Determinants

Notations

Definition of a nth Order Determinant

Minors, Cofactors

Laplacian Development of Determinants by a Row (or a Column)

Properties of Determinants

Cramer's Rule

Nonhomogeneous Systems

Homogeneous Systems

Block Diagonal Determinants

Laplacian Developments by Complementary Minors

Multiplication of Determinants of the Same Order

Differentiation of Determinants

Determinants in Geometry

Exercises

Matrix Algebra

Matrix Notation

Definition

Some Special Matrices

Matrix Equation

Transpose of a Matrix

Matrix Multiplication

Product of Two Matrices

Motivation of Matrix Multiplication

Properties of Product Matrices

Determinant of Matrix Product

The Commutator

Systems of Linear Equations

Gauss Elimination Method

Existence and Uniqueness of Solutions of Linear Systems

Inverse Matrix

Nonsingular Matrix

Inverse Matrix by Cramer's Rule

Inverse of Elementary Matrices

Inverse Matrix by Gauss-Jordan Elimination

Exercises

Eigenvalue Problems of Matrices

Eigenvalues and Eigenvectors

Secular Equation

Properties of Characteristic Polynomial

Properties of Eigenvalues

Some Terminology

Hermitian Conjugation

Orthogonality

Gram-Schmidt Process

Unitary Matrix and Orthogonal Matrix

Unitary Matrix

Properties of Unitary Matrix

Orthogonal Matrix

Independent Elements of an Orthogonal Matrix

Orthogonal Transformation and Rotation Matrix

Diagonalization

Similarity Transformation

Diagonalizing a Square Matrix

Quadratic Forms

Hermitian Matrix and Symmetric Matrix

Definitions

Eigenvalues of Hermitian Matrix

Diagonalizing a Hermitian Matrix

Simultaneous Diagonalization

Normal Matrix

Functions of a Matrix

Polynomial Functions of a Matrix

Evaluating Matrix Functions by Diagonalization

The Cayley-Hamilton Theorem

Exercises

References

Index

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