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