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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 | |