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"Mathematics Formulary by ir. J.C.A. Wevers (c) 1999, 2002 J.C.A. Wevers Version: January 6, 2002 Dear reader, This document contains 66 pages with mathematical equations intended for physicists and engineers. It is a short reference for anyone who often needs to look up mathematical equations. The document can be obtained from the author, Johan Wevers (johanw@xs4all.nl) or found on the WWW at http://www.xs4all.nl/~johanw/index.html. This document is Copyright by J.C.A. Wevers. All rights reserved. Permission to use, copy and distribute this unmodified document by any means and for any purpose except profit purposes is hereby granted. Reproducing this document by any means, including but not limited to printing, copying existing prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause unless upon explicit prior written permission of the author. The C code for rootfinding via Newton's method and the FFT in chapter 8 are from "Numerical Recipes in C", 2nd Edition, ISBN 0-521-43108-5. The Mathematics Formulary is made with TeX and eTeX version 2.09. If you prefer the notation where vectors are typeset in boldface, uncomment the redefinition of the vec command and recompile the file. If you find any errors or have comments, please let me know. I am always open for suggestions and possible corrections to the mathematics formulary. Johan Wevers Contents 1 Basics 1.1 Goniometric functions 1.2 Hyperbolic functions 1.3 Calculus 1.4 Limits 1.5 Complex numbers and quaternions 1.5.1 Complex numbers 1.5.2 Quaternions 1.6 Geometry 1.6.1 Triangles 1.6.2 Curves 1.7 Vectors 1.8 Series 1.8.1 Expansion 1.8.2 Convergence and divergence of series 1.8.3 Convergence and divergence of functions 1.9 Products and quotients 1.10 Logarithms 1.11 Polynomials 1.12 Primes 2 Probability and statistics 2.1 Combinations 2.2 Probability theory 2.3 Statistics 2.3.1 General 2.3.2 Distributions 2.4 Regression analyses 3 Calculus 2 3.1 Integrals 3.1.1 Arithmetic rules 3.1.2 Arc lengths, surfaces and volumes 3.1.3 Separation of quotients 3.1.4 Special functions 3.1.5 Goniometric integrals 3.2 Functions with more variables 3.2.1 Derivatives 3.2.2 Taylor series 3.2.3 Extrema 3.2.4 The V-operator 3.2.5 Integral theorems 3.2.6 Multiple integrals 3.2.7 Coordinate transformations 3.3 Orthogonality of functions 3.4 Fourier series 4 Differential equations 4.1 Linear differential equations 4.1.1 First order linear DE 4.1.2 Second order linear DE 4.1.3 The Wronskian 4.1.4 Power series substitution 4.2 Some special cases 4.2.1 Frobenius method 4.2.2 Euler 4.2.3 Legendre's DE 4.2.4 The associated Legendre equation 4.2.5 Solutions for Bessel's equation 4.2.6 Properties of Bessel functions 4.2.7 Laguerre's equation 4.2.8 The associated Laguerre equation 4.2.9 Hermite 4.2.10 Chebyshev 4.2.11 Weber 4.3 Non-linear differential equations 4.4 Sturm-Liouville equations 4.5 Linear partial differential equations 4.5.1 General 4.5.2 Special cases 4.5.3 Potential theory and Green's theorem 5 Linear algebra 5.1 Vector spaces 5.2 Basis 5.3 Matrix calculus 5.3.1 Basic operations 5.3.2 Matrix equations 5.4 Linear transformations 5.5 Plane and line 5.6 Coordinate transformations 5.7 Eigenvalues 5.8 Transformation types 5.9 Homogeneous coordinates 5.10 Inner product spaces 5.11 The Laplace transformation 5.12 The convolution 5.13 Systems of linear differential equations 5.14 Quadratic forms 5.14.1 Quadratic forms in IR2 5.14.2 Quadratic surfaces in IR3 6 Complex function theory 6.1 Functions of complex variables 6.2 Complex integration 6.2.1 Cauchy's integral formula 6.2.2 Residue 6.3 Analytical functions defined by series 6.4 Laurent series 6.5 Jordan's theorem" Ключевые слова: coordinate, cylinder, projection, parameter curve, normal vector, boundary, tan, diffusion equation, derivative, analytical, linear transformation, mathematics, define, laplace transformation, dx, invariant, error, binomial distribution, column rank, rule, commutative, probability, space, lde, curvature, integral, spherical coordinate, transform, fourier series, wave equation, hyperbolic, invariant subspace, calculus, cartesian coordinate, parameter equation, xn, equation, main theorem, gamma function, continuous function, inverse, column, rank, general solution, det, chapter, transformation, oo, finite number, general, dz, mersenne number, harmonic function, taylor series, dimension, independent, continuous, v v, iteration, operator, mathematics forrriulary ir, plane, vector, jordan curve, ip, convergence, product space, constant coefficient, osculation plane, homogeneous, term, equal, laurent series, wevers, positive direction, distribution, forrriulary, definition, arc length, linear, big circle, root, case, log, commuting, tangent plane, exp, dy, total number, riemann tensor, algorithm requires, orthogonal transformation, formula, symmetric, homogeneous equation, fourier transform, circle, differential, method, sin, linear independent, form, outer product, mathematics formulary, complex, prime, hfixn, bessels equation, number, base, polynomial, fourier, root root, unique solution, success, point, regular point, solution, weners, length, polar coordinate, constant, xe, ha, zn, coordinate transformation, document, curve, differential operator, fundamental tensor, fermat number, possibility, simple pole, unitary, hermitian, mathematics forrriulary, primitive function, singular, principal, infinite number, dt, gauss elimination, tangent, ai, theorem, ax, volume, row, dy dz, order linear, vector field, vector space, written, du, christoffel symbol, limiting case, directional derivative, relation hold, laplace transform, element, linear subspace, newtons method, notation, representation, weight function, xl, isometric transformation, iim, distribution describing, function, normal transformation, initial condition, set, ir, standard deviation, common eigenvectors, uniform convergent, matrix, suppose, points, basis, product, order, normal derivative, normal curvature, integral formula, orthonormal, linear combination, mutually perpendicular, defined, series, characteristic equation, eigenvalue, normal, convergent, orthogonal, df, ellipse, differential equation, complex number, tensor, fixed, float, small circle, space curve, data, subspace, delta function, exists, property, double, surface, christojfel symbol, hold