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Spiegel M.R. — Mathematical Handbook of Formulas and Tables
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Название: Mathematical Handbook of Formulas and TablesАвтор: Spiegel M.R. Аннотация: Book Description
Язык: Рубрика: Математика /Справочники /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1968Количество страниц: 271Добавлена в каталог: 21.10.2004Операции: Положить на полку |
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Equation of plane, general 47 Equation of plane, intercept form for 47 Equation of plane, normal form for 48 Equation of plane, passing through three points 47 Error function 183 Error function, complementary 183 Error function, table of values of 257 Euler numbers 114 115 Euler numbers, definition of 114 Euler numbers, relationship of, to Bernoulli numbers 115 Euler numbers, series involving 115 Euler numbers, table of first few 114 Euler or Cauchy differential equation 105 Euler — Maclaurin summation formula 109 Euler's constant 1 Euler's identities 24 Evolute of an ellipse 44 Exact differential equation 104 Exponential functions 23—25 200 Exponential functions, periodicity of 24 Exponential functions, relationship of to trigonometric functions 24 Exponential functions, sample problems involving calculation of 200 Exponential functions, series for 111 Exponential functions, table of 226 227 Exponential integral 183 Exponential integral, table of values for 251 Exponents 23 F distribution 189 F distribution, 95th and 99th percentile values for 260 261 Factorial n 3 Factorial n, table of values for 234 Factors 2 Focus of conic 37 Focus of ellipse 38 Focus of hyperbola 39 Focus of parabola 38 Folium of Descartes 43 Fourier series 131—135 Fourier series, complex form of 131 Fourier series, convergence of 131 Fourier series, definition of 131 Fourier series, Parseval's identity for 131 Fourier series, special 132—135 Fourier transforms 174—178 Fourier transforms sine 175 Fourier transforms, convolution theorem for 175 Fourier transforms, cosine 176 Fourier transforms, definition of 175 Fourier transforms, Parseval's identity for 175 Fourier transforms, table of 176—178 Fourier's integral theorem 174 Fresnel sine and cosine integrals 184 Frullani's integral 100 Frustrum of right circular cone, lateral surface area of 9 Frustrum of right circular cone, volume of 9 Gamma function 1 101 102 Gamma function for negative values 101 Gamma function, asymptotic expansions for 102 Gamma function, definition of 101 102 Gamma function, derivatives of 102 Gamma function, duplication formula for 102 Gamma function, graph of 101 Gamma function, infinite product for 102 188 Gamma function, recursion formula for 101 Gamma function, relationship of to beta function 103 Gamma function, relationships involving 102 Gamma function, special values for 101 Gamma function, table of values for 235 Gauss' theorem 123 Gaussian plane 22 Generalized integration by parts 59 Generating functions 137 139 146 149 151 153 155 157 158 Geometric formulas 5—10 Geometric mean 185 Geometric series 107 Geometric series, arithmetic- 107 Gradient 119 Gradient in curvilinear coordinates 125 Green's first and second identities 124 Green's theorem 123 Half angle for trigonometric functions 16 Half angle formulas for hyperbolic functions 27 Half rectified sine wave function 172 Hankel functions 138 Harmonic mean 185 Heaviside's unit function 173 Hermite polynomials 151 152 Hermite polynomials, addition formulas for 152 Hermite polynomials, generating function for 151 Hermite polynomials, orthogonal series for 152 Hermite polynomials, orthogonality of 152 Hermite polynomials, recurrence formulas for 151 Hermite polynomials, Rodrigue's formula for 151 Hermite polynomials, special 151 Hermite polynomials, special results involving 152 Hermite's differential equation 151 Higher derivatives 55 Higher derivatives, Leibnitz rule for 55 Holder's inequality 185 Holder's inequality for integrals 186 Homogeneous differential equation 104 Homogeneous differential equation, linear second order 105 Hyperbola 37 39 Hyperbola, asymptotes of 39 Hyperbola, eccentricity of 39 Hyperbola, equation of 37 Hyperbola, focus of 39 Hyperbola, length of major and minor axes of 39 Hyperbolas, confocal 127 Hyperbolic functions 26—31 Hyperbolic functions of negative arguments 26 Hyperbolic functions, addition formulas for 27 Hyperbolic functions, definition of 26 Hyperbolic functions, double angle formulas for 27 Hyperbolic functions, graphs of 29 Hyperbolic functions, half angle formulas for 27 Hyperbolic functions, multiple angle formulas for 27 Hyperbolic functions, periodicity of 31 Hyperbolic functions, powers of 28 Hyperbolic functions, relationship of to trigonometric functions 31 Hyperbolic functions, relationships among 26 28 Hyperbolic functions, sample problems for calculation of 200 201 Hyperbolic functions, series for 112 Hyperbolic functions, sum, difference and product of 28 Hyperbolic functions, table of values for 228—233 Hyperbolic paraboloid 52 Hyperboloid of one sheet 51 Hyperboloid of two sheets 52 Hypergeometric differential distribution 189 Hypergeometric differential equation 160 Hypergeometric functions 160 Hypergeometric functions, miscellaneous properties of 160 Hypergeometric functions, special cases of 160 Hypocycloid with four cusps 40 Hypocycloid, general 42 Imaginary part of a complex number 21 Imaginary unit 21 Improper integrals 94 Indefinite integrals 57—93 Indefinite integrals, definition of 57 Indefinite integrals, table of 60—93 Indefinite integrals, transformation of 59 60 Inequalities 185 186 Infinite products 102 188 Infinite series see “Series” Initial point of a vector 116 Integral calculus, fundamental theorem of 94 Integrals double 122 Integrals improper 94 Integrals multiple 122 125 Integrals, involving vectors 121 Integration 57 (see also “Integrals”) Integration by parts 57 Integration by parts, generalized 59 Integration, constants of 57 Integration, general rules of 57—59 Intercepts 34 47 Interest 201 240—243 Interpolation 195 Interval of convergence 110 Inverse hyperbolic functions 29—31 Inverse hyperbolic functions, definition of 29 Inverse hyperbolic functions, expressed in terms of logarithmic functions 29 Inverse hyperbolic functions, graphs of 30 Inverse hyperbolic functions, principal values for 29 Inverse hyperbolic functions, relationship of to inverse trigonometric functions 31 Inverse hyperbolic functions, relationships between 30 Inverse Laplace transforms 161 Inverse trigonometric functions 17—19 Inverse trigonometric functions, definition of 17 Inverse trigonometric functions, graphs of 18 19 Inverse trigonometric functions, principal values for 17 Inverse trigonometric functions, relations between 18 Inverse trigonometric functions, relationship of to inverse hyperbolic functions 31 Involute of a circle 43 Jacobi's elliptic functions 180 Jacobian 125 Ker and Kei functions 140 141 Ker and Kei functions, definition of 140 Ker and Kei functions, differential equation for 141 Ker and Kei functions, graphs of 141 Lagrange form of remainder in Taylor series 110 Laguerre polynomials 153 154 Laguerre polynomials, generating function for 153 Laguerre polynomials, orthogonal series for 154 Laguerre polynomials, orthogonality of 154 Laguerre polynomials, recurrence formulas for 153 Laguerre polynomials, Rodrigue's formula for 153 Laguerre polynomials, special 153 Laguerre's associated differential equation 155 Laguerre's differential equation 153 Landen's transformation 180 Laplace transforms 161—173 Laplace transforms, complex inversion formula for 161 Laplace transforms, definition of 161 Laplace transforms, inverse 161 Laplace transformsm, table of 162—173 Laplacian 120 Laplacian in curvilinear coordinates 125 Legendre functions 146—148 (see also “Legendre polynomials”) Legendre functions of the second kind 148 Legendre polynomials 146 147 Legendre polynomials, generating function for 146 Legendre polynomials, orthogonal series of 147 Legendre polynomials, orthogonality of 147 Legendre polynomials, recurrence formulas for 147 Legendre polynomials, Rodrigue's formula for 146 Legendre polynomials, special 146 Legendre polynomials, special results involving 147 Legendre polynomials, table of values for 252 253 Legendre's associated differential equation 149 Legendre's associated differential equation, general solution of 150 Legendre's differential equation 106 146 Legendre's differential equation, general solution of 148 Legendre's relation for elliptic integrals 182 Leibnitz's rule for differentiation of integrals 95 Leibnitz's rule for higher derivatives of products 55 Lemniscate 40 44 Limacon of Pascal 41 44 Line integrals 121 122 Line integrals, definition of 121 Line integrals, independence of path of 121 122 Line integrals, properties of 121 Line, slope of 34 Linear first order differential equation 104 Linear second order differential equation 105 Logarithmic functions 23—25 (see also “Logarithms”) Logarithmic functions, series for 111 Logarithms 23 (see also “Logarithmic functions”) Logarithms of complex numbers 25 Logarithms of trigonometric functions 216—221 Logarithms, base of 23 Logarithms, Briggsian 23 Logarithms, change of base of 24 Logarithms, characteristic of 194 Logarithms, mantissa of 194 Logarithms, natural 24 Maclaurin series 110 Mantissa 194 Mean value theorem for definite integrals 94 Mean value theorem, generalized 95 Minkowski's inequality 186 Minkowski's inequality for integrals 186 Modified Bessel functions 138 139 Modified Bessel functions of order half an odd integer 140 Modified Bessel functions recurrence formulas for 139 Modified Bessel functions, differential equation for 138 Modified Bessel functions, generating function for 139 Modified Bessel functions, graphs of 141 Modulus of a complex number 22 Moments of inertia special 190 191 Multinomial formula 4 Multiple angle formulas for hyperbolic functions 27 Multiple angle formulas for trigonometric functions 16 Multiple integrals 122 Multiple integrals, transformation of 125 Napier's rules 20 Napierian logarithms 24 196 Napierian logarithms, tables of 224 225 Natural logarithms and antilogarithms 24 196 Natural logarithms and antilogarithms, tables of 224—227 Neumann's function 136 Nonhomogeneous equation, linear second order 105 Normal curve, areas under 257 Normal curve, ordinates of 256 Normal distribution 189 Normal form, equation of line in 35 Normal form, equation of plane in 48 Normal outward drawn or positive 123 Normal unit 122 Null function 170 Null vector 116 Oblate spheroidal coordinates 128 Oblate spheroidal coordinates, Laplacian in 128 Orthogonal curvilinear coordinates 124—130 Orthogonal curvilinear coordinates, formulas involving 125 Orthogonality and orthogonal series 144 145 147 150 152 154 156 158 159 Ovals of Cassini 44 parabola 37 38 Parabola, eccentricity of 37 Parabola, equation of 37 38 Parabola, focus of 38 Parabolas, confocal 126 Parabolic cylindrical coordinates 126 Parabolic cylindrical coordinates, Laplacian in 126 Parabolic formula for definite integrals 95 Paraboloid of revolution, volume of 10 Paraboloid, elliptic 52 Paraboloid, hyperbolic 52 Paraboloidal coordinates 127 Paraboloidal coordinates, Laplacian in 127 Parallel, condition for lines to be 35 Parallelepiped, volume of 8 Parallelogram law for vector addition 116 Parallelogram, area of 5 Parallelogram, perimeter of 5 Parseval's identity for Fourier series 131 Parseval's identity for Fourier transforms 175 Partial derivatives 56 Partial fraction expansions 187
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