Applied Mathematics for Engineers and Physicists by Pipes, Louis A. -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Preface to The Dover Edition Preface Contents Chapter 1 The Theory of Complex Variables

1 Introduction 2 Functions of a Complex Variable 3 The Derivative and the Cauchy-Riemann Differential Equations 4 Line Integrals of Complex Functions 5 Cauchy’s Integral Theorem 6 Cauchy’s Integral Formula 7 Taylor’s Series 8 Laurent’s Series 9 Residues: Cauchy’s Residue Theorem 10 Singular Points of an Analytic Function 11 The Point at Infinity 12 Evaluation of Residues 13 Liouville’s Theorem 14 Evaluation of Definite Integrals 15 Jordan’s Lemma 16 Bromwich Contour Integrals 17 Integrals Involving Multiple-valued Functions (Branch Points) 18 Further Examples of Contour Integrals Around Branch Points 19 The Use of z and in the Theory of Complex Variables Problems References

Chapter 2 Linear Differential Equations

1 Introduction 2 The Reduced Equation; the Complementary Function 3 Properties of the Operator Ln(D) 4 The Method of Partial Fractions 5 Linear Dependence: Wronskian 6 The Method of Undetermined Coefficients 7 The Use of Complex Numbers to Find the Particular Integral 8 Linear Second-order Differential Equations with variable Coefficients 9 The Method of Frobenius 10 Variation of Parameters 11 The Sturm-Liouville Differential Equation Problems References

Chapter 3 Linear Algebraic Equations, Determinants, and Matrices

1 Introduction 2 Simple Determinants 3 Fundamental Definitions 4 Laplace Expansion 5 Fundamental Properties of Determinants 6 The Evaluation of Numerical Determinants 7 Definition of a Matrix 8 Special Matrices 9 Equality of Matrices; Addition and Subtraction 10 Multiplication of Matrices 11 Matrix Division, the Inverse Matrix 12 The Reversal Law in Transposed and Reciprocated Products 13 Properties of Diagonal and Unit Matrices 14 Matrices Partitioned into Submatrices 15 Matrices of Special Types 16 The Solution of Linear Algebraic Equations 17 The Special Case of n Equations and n Unknowns 18 Systems of Homogeneous Linear Equations 19 The Characteristic Matrix and the Characteristic Equation of a Matrix 20 Eigenvalues and the Reduction of a Matrix to Diagonal Form 21 The Trace of a Matrix 22 The Cayley-Hamilton Theorem 23 The Inversion of Large Matrices 24 Sylvester’s Theorem 25 Power Series of Matrices; Functions of Matrices 26 Alternate Method of Evaluating Functions of Matrices 27 Differentiation and Integration of Matrices 28 Association of Matrices with Linear Differential Equations 29 Method of Peano-Baker 30 Adjoint Method 31 Existence and Uniqueness of Solutions of Matrix Differential Equations 32 Linear Equations with Periodic Coefficients 33 Matrix Solution of the Hill-Meissner Equation 34 The Use of Matrices to Determine the Roots of Algebraic Equations Problems References

Chapter 4 Laplace Transforms

1 Introduction 2 The Fourier-Mellin Theorem 3 The Fundamental Rules 4 Calculation of Direct Transforms 5 Calculation of Inverse Transforms 6 The Modified Integral 7 Impulsive Functions 8 Heaviside’s Rules 9 The Transforms of Periodic Functions 10 The Simple Direct Laplace-transform, or Operational, Method of Solving Linear Differential Equations with Constant Coefficients 11 Systems of Linear Differential Equations with Constant Coefficients Problems References

Chapter 5 Oscillations of Linear Lumped Electrical Circuits

1 Introduction 2 Electrical-circuit Principles 3 Energy Considerations 4 Analysis of General Series Circuit 5 Discharge and Charge of a Capacitor 6 Circuit with Mutual Inductance 7 Circuits Coupled by a Capacitor 8 The Effect of Finite Potential Pulses 9 Analysis of the General Network 10 The Steady-state Solution 11 Four-terminal Networks in the Alternating-current Steady State 12 The Transmission Line as a Four-terminal Network Problems References

Chapter 6 Oscillations of Linear Mechanical Systems

1 Introduction 2 Oscillating Systems with One Degree of Freedom 3 Two Degrees of Freedom 4 Lagrange’s Equations 5 Proof of Lagrange’s Equations 6 Small Oscillations of Conservative Systems 7 Solution of the Frequency Equation and Calculation of the Normal Modes by the Use of Matrices 8 Numerical Example: the Triple Pendulum 9 Nonconservative Systems: Vibrations with Viscous Damping 10 A Matrix Iterative Method for the Analysis of Nonconservative Systems 11 Forced Oscillations of a Nonconservative System Problems References

Chapter 7 The Calculus of Finite Differences and Linear Difference Equations With Constant Coefficients

1 Introduction 2 The Fundamental Operators of the Calculus of Finite Differences 3 The Algebra of Operators 4 Fundamental Equations Satisfied by the Operators 5 Difference Tables 6 The Gregory-Newton Interpolation Formula 7 The Derivative of a Tabulated Function 8 The Integral of a Tabulated Function 9 A Summation Formula 10 Difference Equation with Constant Coefficients 11 Oscillations of a Chain of Particles Connected by Strings 12 An Electrical Line with Discontinuous Leaks 13 Filter Circuits 14 Four-terminal-network Connection with Matrix Algebra 15 Natural Frequencies of the Longitudinal Motions of Trains Problems References

Chapter 8 Transfer Functions and Impulse Responses

1 Introduction 2 Transfer Functions of Linear Systems 3 Solutions to Problems Using Transfer Functions 4 Combining Transfer Functions of Several Systems 5 Matrix Method for Evaluating Over-all Transfer Functions When Loading Occurs 6 Method 7 Impulse Responses and Transfer Functions 8 Feedback Control in Linear Systems 9 Stability of Linear Systems Problems References

Chapter 9 Laplace’s Equation

1 Introduction 2 Laplace’s Equation in Cartesian, Cylindrical, and Spherical Coordinate Systems 3 Two-dimensional Steady Flow of Heat 4 Circular Harmonics 5 Conducting Cylinder in a Uniform Field 6 General Cylindrical Harmonics 7 Spherical Harmonics 8 The Potential of a Ring 9 The Potential about a Spherical Surface 10 General Properties of Harmonic Functions Problems References

Chapter 10 The Solution of Two-Dimensional Potential Problems by The Method of Conjugate Functions

1 Introduction 2 Conjugate Functions 3 Conformal Representation 4 Basic Principles of Electrostatics 5 The Transformation z = k cosh w 6 General Powers of z 7 The Transformation 8 Determination of the Required Transformation When the Boundary Is Expressed in Parametric Form 9 Schwartz’s Transformation 10Polygon with One Angle 11 Successive Transformations 12 The Parallel-plate Capacitor; Flow Out of a Channel 13 The Effect of a Wall on a Uniform Field 14 Application to Hydrodynamics 15 Application to Steady Heat Flow Problems References

Chapter 11 The Equation of Heat Conduction or Diffusion

1 Introduction 2 Variable Linear Flow 3 Electrical Analogy of Linear Heat Flow 4 Linear Flow in Semi-infinite Solid, Temperature on Face Given as Sinusoidal Function of Time 5 Two-dimensional Heat Conduction 6 Temperatures in an Infinite Bar 7 Temperatures Inside a Circular Plate 8 Skin Effect on a Plane Surface 9 Current Density in a Wire 10 General Theorems Problems Reference

Chapter 12 The Wave Equation

1 Introduction 2 The Transverse Vibrations of a Stretched String 3 D’Alembert’s Solution; Waves on Strings 4 Harmonic Waves 5 Fourier-series Solution 6 Orthogonal Functions 7 The Oscillations of a Hanging Chain 8 The Vibrations of a Rectangular Membrane 9 The Vibrations of a Circular Membrane 10 The Telegraphist’s, or Transmission-line, Equations 11 Tidal Waves in a Canal 12 Sound Waves in a Gas 13 The Magnetic Vector Potential 14 The Inhomogeneous Wave Equation 15 The Theory of Waveguides Problems References

Chapter 13 Operational Methods in Applied Mathematics

1 Introduction 2 Integral Transforms 3 Application of the Operational Calculus to the Solution of Partial Differential Equations 4 Evaluation of Integrals 5 Applications of the Laplace Transform to the Solution of Linear Integral Equations 6 Solution of Ordinary Differential Equations with Variable Coefficients 7 The Summation of Fourier Series by the Laplace Transform 8 The Deflection of a Loaded Cord 9 Stretched Cord with Elastic Support 10 The Deflection of Beams by Transverse Forces 11 Deflection of Beams on an Elastic Foundation 12 Buckling of a Uniform Column under Axial Load 13 The Vibration of Beams 14 Examples of Fourier Transforms 15 Mellin and Hankel Transforms 16 Repeated Use of Transforms 17 Green’s Functions Problems References

Chapter 14 Approximate Methods in Applied Mathematics

1 Introduction 2 The Method of Least Squares 3 Matrix Formulation of the Method of Least Squares 4 Numerical Integration of First-order Differential Equations 5 Numerical Integration of Higher-order Differentia Equations 6 Monte Carlo Method 7 Approximate Solutions of Differential Equations 8 Rayleigh’s Method of Calculating Natural Frequencies 9 The Collocation Method 10 The Method of Rayleigh-Ritz 11 Galerkin’s Method Problems References

Chapter 15 The Analysis of Nonlinear Systems

1 Introduction 2 Oscillator Damped by Solid Friction 3 The Free Oscillations of a Pendulum 4 Restoring Force a General Function of the Displacement 5 An Operational Analysis of Nonlinear Dynamical Systems 6 Forced Vibrations of Nonlinear Systems 7 Forced Oscillations with Damping 8 Solution of Nonlinear Differential Equations by Integral Equations 9 The Method of Kryloff and Bogoliuboff 10 Applications of the Method of Kryloff and Bogoliuboff 11 Topological Methods: Autonomous Systems 12 Nonlinear Conservative Systems 13 Relaxation Oscillations 14 Phase Trajectories of the van der Pol Equations 15 The Period of Relaxation Oscillations 16 Relaxation Oscillations of a Motor-Generator Combination The Reversion Method for Solving Nonlinear Differential Equations 17 Introduction 18 General Description of the Method 19 Examples Illustrating the Method 20 Conclusion Forced Oscillations of Nonlinear Circuits 21 Introduction 22 Forced Oscillations of a Nonlinear Inductor 23 Oscillations of a Saturable Reactor 24 Forced Oscillations of a Nonlinear Capacitor 25 Steady-state Oscillations of a Series-connected Magnetic Amplifier 26 Conclusions Matrix Solution of Equations of the Mathieu-Hill Type 27 Introduction 28 The Use of Matrix Algebra in Solving Hill’s Equation 29 The Solution of the Hill-Meissner Equation 30 Solution of Hill’s Equation if F(t) Is a Sum of Step Functions 31 A Class of Hill’s Equations with Exponential Variation 32 Hill’s Equation with a Sawtooth Variation 33 Conclusion The Analysis of Time-varying Electrical Circuits 34 Introduction 35 The Classical Theory of Differential Equations 36 Matrix Methods in the Analysis of Time-variable Circuits 37 Approximate Solution of Time-variable Circuit Problems by the Use of the BWK Approximation 38 The Use of Laplace Transforms and Integral Equations in the Solution of Time-variable Circuit Problems 39 Conclusion Analysis of Linear Time-varying Circuits by the Brillouin-Wentzel-Kramers Method 40 Introduction 41 BWK Approximation 42 Capacitance Modulation 43 Forced Oscillations of a Circuit with Variable Elastance 44 Series Circuit with Periodically Varying Resistance 45 Series Circuit with Periodically Varying Inductance 46 Conclusion Problems References

Chapter 16 Statistics and Probability

1 Introduction 2 Statistical Distributions 3 Second Moments and Standard Deviation 4 Definitions of Probability 5 Fundamental Laws of Probability 6 Discrete Probability Distributions 7 Elements of the Theory of Combinations and Permutations 8 Stirling’s Approximation for the Factorial 9 Continuous Distributions 10 Expectation, Moments, and Standard Deviation 11 The Binomial Distribution 12 The Poisson Distribution 13 The Normal or Gaussian Distribution 14 Distribution of a Sum of Normal Variates 15 Applications to Experimental Measurements 16 The Standard Deviation of the Mean Problems References

Appendix A Table of Laplace Transforms Appendix B Special Functions of Applied Mathematics

1 Introduction 2 Bessel’s Differential Equation 3 Series Solution of Bessel’s Differential Equation 4 The Bessel Function of Order n of the Second Kind 5 Values of Jn(x) and Yn(x) for Large and Small Values of x 6 Recurrence Formulas for Jn(x) 7 Expressions for Jn(x) When n Is Half an Odd Integer 8 The Bessel Functions of Order n of the Third Kind, or Hankel Functions of Order n 9 Differential Equations Whose Solutions Are Expressible in Terms of Bessel Functions 10 Modified Bessel Functions 11 The ber and bei Functions 12 Expansion in Series of Bessel Functions 13 The Bessel Coefficients 14 Legendre’s Differential Equation 15 Rodrigues’ Formula for the Legendre Polynomials 16 Legendre’s Function of the Second Kind 17 The Generating Function for Pn(x) 18 The Legendre Coefficients 19 The Orthogonality of Pn(x) 20 Expansion of an Arbitrary Function in a Series of Legendre Polynomials 21 Associated Legendre Polynomials 22 The Gamma Function 23 The Factorial; Gauss’s Pi Function 24 The Value of Γ(½); Graph of the Gamma Function 25 The Beta Function 26 The Connection of the Beta Function and the Gamma Function 27 An Important Relation Involving Gamma Functions 28 The Error Function or Probability Integral Problems References

Appendix C Infinite Series, Fourier Series, and Fourier Integrals

1 Infinite Series 2 Definitions 3 The Geometric Series 4 Convergent and Divergent Series 5 General Theorems 6 The Comparison Test 7 Cauchy’s Integral Test 8 Cauchy’s Ratio Test 9 Alternating Series 10 Absolute Convergence 11 Power Series 12 Theorems Regarding Power Series 13 Series of Functions and Uniform Convergence 14 Integration and Differentiation of Series 15 Taylor’s Series 16 Symbolic Form of Taylor’s Series 17 Evaluation of Integrals by Means of Power Series 18 Approximate Formulas Derived from Maclaurin’s Series 19 Use of Series for the Computation of Functions 20 Evaluation of a Function Taking on an Indeterminate Form 21 Fourier Series and Integrals 22 Representation of More Complicated Periodic Phenomena; Fourier Series 23 Examples of Fourier Expansions of Functions 24 Some Remarks about Convergence of Fourier Series 25 Effective Values and the Average of a Product 26 Modulated Vibrations and Beats 27 The Propagation of Periodic Disturbances in the Form of Waves 28 The Fourier Integral Problems References

Appendix D The Solution of Transcendental and Polynomial Equations

1 Introduction 2 Graphical Solution of Transcendental Equations 3 The Newton-Raphson Method 4 Solution of Cubic Equations 5 Graffe’s Root-squaring Method Problems References

Appendix E Vector and Tensor Analysis

1 Introduction 2 The Concept of a Vector 3 Addition and Subtraction of Vectors; Multiplication of a Vector by a Scalar 4 The Scalar Product of Two Vectors 5 The Vector Product of Two Vectors 6 Multiple Products 7 Differentiation of a Vector with Respect to Time 8 The Gradient 9 The Divergence and Gauss’s Theorem 10 The Curl of a Vector Field and Stokes’s Theorem 11 Successive Applications of the Operator A 12 Orthogonal Curvilinear Coordinates 13 Application to Hydrodynamics 14 The Equation of Heat Flow in Solids 15 The Gravitational Potential 16 Maxwell’s Equations 17 The Wave Equation 18 The Skin-effect, or Diffusion, Equation 19 Tensors (Qualitative Introduction) 20 Coordinate Transformations 21 Scalars, Contravariant Vectors, and Covariant Vectors 22 Addition, Multiplication, and Contraction of Tensors 23 Associated Tensors 24 Differentiation of an Invariant 25 Differentiation of Tensors: The Christoffel Symbols 26 Intrinsic and Covariant Derivatives of Tensors of Higher Order 27 Application of Tensor Analysis to the Dynamics of a Particle Problems References

Appendix F Partial Differentiation and The Calculus of Variations

1 Introduction 2 Partial Derivatives 3 The Symbolic Form of Taylor’s Expansion 4 Differentiation of Composite Functions 5 Change of Variables 6 The First Differential 7 Differentiation of Implicit Functions 8 Maxima and Minima 9 Differentiation of a Definite Integral 10 Integration under the Integral Sign 11 Evaluation of Certain Definite Integrals 12 The Elements of the Calculus of Variations 13 Summary of Fundamental Formulas of the Calculus of Variations 14 Hamilton’s Principle; Lagrange’s Equations 15 Variational Problems with Accessory Conditions: Isoperimetric Problems Problems References

Appendix G Answers to Selected Problems Index

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