Review of Fundamental Concepts                 1  (37)
      Real numbers
      Rules of algebra
      Functions
      Special types of functions
      Limits
      Continuity
      Derivatives
      Differentiation formulas
      Integrals
      Integration formulas
      Sequences and series
      Uniform convergence
      Taylor series
      Functions of two or more variables
      Partial derviatives
      Taylor series for functions of two or
      more variables
      Linear equations and determinants
      Maxima and minima
      Method of Lagrange multipliers
      Leibnitz's rule for differentiating an
      integral
      Multiple integrals
      Complex numbers
    Ordinary Differential Equations                38 (33)
      Definition of a differential equation
      Order of a differential equation
      Arbitrary contants
      Solution of a differential equation
      Differential equation of a family of
      curves
      Special first order equations and
      solutions
      Equations of higher order
      Existence and uniqueness of solutions
      Applications of differential equations
      Some special applications
      Mechanics
      Electric circuits
      Orthogonal trajectories
      Deflection of beams
      Miscellaneous problems
      Numerical methods for solving
      differential equations
    Linear Differential Equations                  71 (27)
      General linear differential equation of
      order n.
      Existence and uniqueness theorem
      Operator notation
      Linear operators
      Fundamental theorem on linear
      differential equations
      Linear dependence and Wronskians
      Solutions of linear equations with
      constant coefficients
      Non-operator techniques
      The complementary or homogeneous solution
      The particular solution
      Method of undetermined coefficients
      Method of variation of parameters
      Operator techniques
      Method of reduction of order
      Method of inverse operators
      Linear equations with variable
      coefficients
      Simultaneous differential equations
      Applications
    Laplace Transforms                             98 (23)
      Definition of a Laplace transform
      Laplace transforms of some elementary
      functions
      Sufficient conditions for existence of
      Laplace transforms
      Inverse Laplace transforms
      Laplace transforms of derivatives
      The unit step function
      Some special theorems on Laplace
      transforms
      Partial fractions
      Solutions of differential equations by
      Laplace transforms
      Applications to physical problems
      Laplace inversion formulas
    Vector Analysis                                121(26)
      Vectors and scalars
      Vector algebra
      Laws of vector algebra
      Unit vectors
      Rectangular unit vectors
      Components of a vector
      Dot or scalar product
      Cross or vector product
      Triple products
      Vector functions
      Limits, continuity and derivative of
      vector functions
      Geometric interpretation of a vector
      derivative
      Gradient, divergence and curl
      Formulas involving Δ
      Orthogonal curvilinear coordinates
      Jacobians
      Gradient, divergence, curl and Laplacian
      in orthogonal curvilinear
      Special curvilinear coordinates
    Multiple, Line and Surface Integrals and       147(35)
    Integral Theorems
      Double integrals
      Iterated integrals
      Triple integrals
      Transformations of multiple integrals
      Line integrals
      Vector notation for line integrals
      Evaluation of line integrals
      Properties of line integrals
      Simple closed curves
      Simply and multiply-connected regions
      Green's theorem in the plane
      Conditions for a line integral to be
      independent of the path
      Surface integrals
      The divergence theorem
      Stokes' theorem
    Fourier Series                                 182(19)
      Periodic functions
      Fourier series
      Dirichlet conditions
      Odd and even functions
      Half range Fourier sine or cosine series
      Parseval's identity
      Differentiation and integration of
      Fourier series
      Complex notation for Fourier series
      Complex notation for Fourier series
      Orthogonal functions
    Fourier Integrals                              201(9)
      The Fourier integral
      Equivalent forms of Fourier's integral
      theorem
      Fourier transforms
      Parseval's identities for Fourier
      integrals
      The convolution theorem
    Gamma, Beta and Other Special Functions        210(14)
      The gamma function
      Table of values and graph of the gamma
      function
      Asymptotic formula for Γ(n)
      Miscellaneous results involving the gamma
      function
      The beta function
      Dirichlet integrals
      Other special functions
      Error function
      Exponential integral
      Sine integral
      Cosine integral
      Fresnel sine integral
      Fresnel cosine integral
      Asymptotic series or expansions
    Bessel Functions                               224(18)
      Bessel's differential equation
      Bessel functions of the first kind
      Bessel functions of the second kind
      Generating functions for Jn(x)
      Recurrence formulas
      Functions related to Bessel functions
      Hankel functions of first and second kinds
      Modified Bessel functions
      Ber, bei, ker, kei functions
      Equations transformed into Bessel's
      equation
      Asymptotic formulas for Bessel functions
      Zeros of Bessel functions
      Orthogonality of Bessel functions
      Series of Bessel functions
    Legendre Functions and Other Orthogonal        242(16)
    Functions
      Legendre's differential equation
      Legendre polynomials
      Generating function for Legendre
      polynomials
      Recurrence formulas
      Legendre functions of the second kind
      Orthogonality of Legendre polynomials
      Series of Legendre polynomials
      Associated Legendre functions
      Other special functions
      Hermite polynomials
      Laguerre polynomials
      Sturm-Lioville systems
    Partial Differential Equations                 258(28)
      Some definitions involving partial
      differential equations
      Linear partial differential equations
      Some important partial differential
      equations
      Heat conduction equation
      Vibrating string equation
      Laplace's equation
      Longitudinal vibrations of a beam
      Transverse vibrations of a beam
      Methods of solving boundary-value problems
      General solutions
      Separation of variables
      Laplace transform methods
    Complex Variables and Conformal Mapping        286(38)
      Functions
      Limits and continuity
      Derivatives
      Cauchy-Riemann equations
      Integrals
      Cauchy's theorem
      Cauchy's integral formulas
      Taylor's series
      Singular points
      Poles
      Laurent's series
      Residues
      Residue theorem
      Evaluation of definite integrals
      Conformal mapping
      Riemann's mapping theorem
      Some general transformations
      Mapping of a half plane on to a circle
      The Schwarz-Christoffel transformation
      Solutions of Laplace's equation by
      conformal mapping
    Complex Inversion Formula for Laplace          324(18)
    Transforms
      The complex inversion formula
      The Bromwich contour
      Use of residue theorem in finding inverse
      Laplace transforms
      A sufficient condition for the integral
      around Γ to approach zero
      Modification of Bromwich contour in case
      of branch points
      Case of infinitely many singularities
      Applications to boundary-value problems
    Matrices                                       342(33)
      Definition of a matrix
      Some special definitions and operations
      involving matrices
      Determinants
      Theorems on determinants
      Inverse of a matrix
      Orthogonal and unitary matrices
      Orthogonal vectors
      Systems of linear equations
      Systems of n equations in n unknowns
      Cramer's rule
      Eigenvalues and eigenvectors
      Theorems on eigenvalues and eigenvectors
    Calculus of Variations                         375(24)
      Maximum or minimum of an integral
      Euler's equation
      Constraints
      The variational notation
      Generalizations
      Hamilton's principle
      Lagrange's equations
      Sturm-Liouville systems and Rayleigh-Ritz
      methods
      Operator interpretation of matrices
Index                                              399