in the Theory of Complex VariablesChapter 1
THE THEORY OF COMPLEX VARIABLES
2 Functions of a Complex Variable
3 The Derivative and the Cauchy-Riemann Differential Equations
4 Line Integrals of Complex Functions
9 Residues: Cauchy’s Residue Theorem
10 Singular Points of an Analytic Function
14 Evaluation of Definite 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
Chapter 2
LINEAR DIFFERENTIAL EQUATIONS
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
11 The Sturm-Liouville Differential Equation
Chapter 3
LINEAR ALGEBRAIC EQUATIONS, DETERMINANTS, AND MATRICES
5 Fundamental Properties of Determinants
6 The Evaluation of Numerical Determinants
9 Equality of Matrices; Addition and Subtraction
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
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
22 The Cayley-Hamilton Theorem
23 The Inversion of Large Matrices
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
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
4 Calculation of Direct Transforms
5 Calculation of Inverse Transforms
9 The Transforms of Periodic Functions
11 Systems of Linear Differential Equations with Constant Coefficients
Chapter 5
OSCILLATIONS OF LINEAR LUMPED ELECTRICAL CIRCUITS
2 Electrical-circuit Principles
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
11 Four-terminal Networks in the Alternating-current Steady State
12 The Transmission Line as a Four-terminal Network
Chapter 6
OSCILLATIONS OF LINEAR MECHANICAL SYSTEMS
2 Oscillating Systems with One Degree of Freedom
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
2 The Fundamental Operators of the Calculus of Finite Differences
4 Fundamental Equations Satisfied by the Operators
6 The Gregory-Newton Interpolation Formula
7 The Derivative of a Tabulated Function
8 The Integral of a Tabulated Function
10 Difference Equation with Constant Coefficients
11 Oscillations of a Chain of Particles Connected by Strings
12 An Electrical Line with Discontinuous Leaks
14 Four-terminal-network Connection with Matrix Algebra
15 Natural Frequencies of the Longitudinal Motions of Trains
Chapter 8
TRANSFER FUNCTIONS AND IMPULSE RESPONSES
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
7 Impulse Responses and Transfer Functions
8 Feedback Control in Linear Systems
2 Laplace’s Equation in Cartesian, Cylindrical, and Spherical Coordinate Systems
3 Two-dimensional Steady Flow of Heat
5 Conducting Cylinder in a Uniform Field
6 General Cylindrical Harmonics
9 The Potential about a Spherical Surface
10 General Properties of Harmonic Functions
Chapter 10
THE SOLUTION OF TWO-DIMENSIONAL POTENTIAL PROBLEMS BY THE METHOD OF CONJUGATE FUNCTIONS
4 Basic Principles of Electrostatics
5 The Transformation z = k cosh w
8 Determination of the Required Transformation When the Boundary Is Expressed in Parametric Form
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
Chapter 11
THE EQUATION OF HEAT CONDUCTION OR DIFFUSION
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
2 The Transverse Vibrations of a Stretched String
3 D’Alembert’s Solution; Waves on Strings
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
13 The Magnetic Vector Potential
14 The Inhomogeneous Wave Equation
Chapter 13
OPERATIONAL METHODS IN APPLIED MATHEMATICS
3 Application of the Operational Calculus to the Solution of Partial Differential Equations
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
14 Examples of Fourier Transforms
15 Mellin and Hankel Transforms
Chapter 14
APPROXIMATE METHODS IN APPLIED MATHEMATICS
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
7 Approximate Solutions of Differential Equations
8 Rayleigh’s Method of Calculating Natural Frequencies
10 The Method of Rayleigh-Ritz
Chapter 15
THE ANALYSIS OF NONLINEAR SYSTEMS
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
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
18 General Description of the Method
19 Examples Illustrating the Method
Forced Oscillations of Nonlinear Circuits
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
Matrix Solution of Equations of the Mathieu-Hill Type
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
The Analysis of Time-varying Electrical Circuits
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
Analysis of Linear Time-varying Circuits by the Brillouin-Wentzel-Kramers Method
43 Forced Oscillations of a Circuit with Variable Elastance
44 Series Circuit with Periodically Varying Resistance
45 Series Circuit with Periodically Varying Inductance
Chapter 16
STATISTICS AND PROBABILITY
3 Second Moments and Standard Deviation
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
10 Expectation, Moments, and Standard Deviation
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
Appendix A
TABLE OF LAPLACE TRANSFORMS
Appendix B
SPECIAL FUNCTIONS OF APPLIED MATHEMATICS
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
12 Expansion in Series of Bessel Functions
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)
20 Expansion of an Arbitrary Function in a Series of Legendre Polynomials
21 Associated Legendre Polynomials
23 The Factorial; Gauss’s Pi Function
24 The Value of Γ(½); Graph of the Gamma 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
Appendix C
INFINITE SERIES, FOURIER SERIES, AND FOURIER INTEGRALS
4 Convergent and Divergent Series
12 Theorems Regarding Power Series
13 Series of Functions and Uniform Convergence
14 Integration and Differentiation of 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
Appendix D
THE SOLUTION OF TRANSCENDENTAL AND POLYNOMIAL EQUATIONS
2 Graphical Solution of Transcendental Equations
5 Graffe’s Root-squaring Method
Appendix E
VECTOR AND TENSOR ANALYSIS
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
7 Differentiation of a Vector with Respect to Time
9 The Divergence and Gauss’s Theorem
10 The Curl of a Vector Field and Stokes’s Theorem
11 Successive Applications of the Operator Δ
12 Orthogonal Curvilinear Coordinates
13 Application to Hydrodynamics
14 The Equation of Heat Flow in Solids
15 The Gravitational Potential
18 The Skin-effect, or Diffusion, Equation
19 Tensors (Qualitative Introduction)
21 Scalars, Contravariant Vectors, and Covariant Vectors
22 Addition, Multiplication, and Contraction of 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
Appendix F
PARTIAL DIFFERENTIATION AND THE CALCULUS OF VARIATIONS
3 The Symbolic Form of Taylor’s Expansion
4 Differentiation of Composite Functions
7 Differentiation of Implicit Functions
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
