Mathematical Methods for Engineers and Scientists 1

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1 K.T. Tang Mathematical Methods for Engineers and Scientists 1 Complex Analysis, Determinants and Matrices With 49 Figures and 2 Tables fyj Springer

2 Part I Complex Analysis 1 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* 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 46

3 VIII Hyperbolic Functions of z Logarithm and General Power of z Inverse Trigonometric and Hyperbolic Functions 55 Exercises 58 2 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 96 Exercises Complex Series and Theory of Residues A Basic Geometric Series Taylor Series The Complex Taylor Series Convergence of Taylor Series Analytic Continuation Ill 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 144

4 3.5.3 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 160 Exercises 165 IX Part II Determinants and Matrices 4 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 204 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 232

5 X 5.3 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 248 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 305 Exercises 309 References 313 Index 315

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS DENNIS G. ZILL Loyola Marymount University MICHAEL R. CULLEN Loyola Marymount University PWS-KENT O I^7 3 PUBLISHING COMPANY E 9 U Boston CONTENTS Preface xiii Parti ORDINARY