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Mathematical Methods for Engineers and Scientists 1 Complex Analysis, Determinants and Matrices

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ISBN-10: 3540302735

ISBN-13: 9783540302735

Edition: 2007

Authors: Kwong-Tin Tang

List price: $99.99
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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
Publication date: 11/10/2006
Binding: Hardcover
Pages: 319
Size: 6.00" wide x 9.25" long x 1.00" tall
Weight: 1.342
Language: English

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