Handbook of mathematical formulas and integrals /

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If there is a formula to solve a given problem in mathematics, you will find it in Alan Jeffrey's Handbook of Mathematical Formulas and Integrals. Thanks to its unique thumb-tab indexing feature, answers are easy to find based upon the type of problem they solve. The Handbook covers important f...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Jeffrey, Alan
Formato: Electrónico eBook
Idioma:Inglés
Publicado: San Diego : Academic Press, �1995.
Temas:
Mathematics > Tables.
Mathematics > Formulae.
Math�ematiques > Tables.
Math�ematiques > Formules.
formulas (algorithms)
Mathematics
tables (documents)
Tables (Data)
Mathematical formulae
Tables
Tables (Donn�ees)
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 0 Quick Reference List of Frequently Used Data
  • 0.1 Useful Identities 1
  • 0.2 Complex Relationships 2
  • 0.3 Constants 2
  • 0.4 Derivatives of Elementary Functions 3
  • 0.5 Rules of Differentiation and Integration 3
  • 0.6 Standard Integrals 4
  • 0.7 Standard Series 11
  • 0.8 Geometry 13
  • 1 Numerical, Algebraic, and Analytical Results for Series and Calculus
  • 1.1 Algebraic Results Involving Real and Complex Numbers 25
  • 1.2 Finite Sums 29
  • 1.3 Bernoulli and Euler Numbers and Polynomials 37
  • 1.4 Determinants 47
  • 1.5 Matrices 55
  • 1.6 Permutations and Combinations 62
  • 1.7 Partial Fraction Decomposition 63
  • 1.8 Convergence of Series 66
  • 1.9 Infinite Products 71
  • 1.10 Functional Series 73
  • 1.11 Power Series 74
  • 1.12 Taylor Series 79
  • 1.13 Fourier Series 81
  • 1.14 Asymptotic Expansions 85
  • 1.15 Basic Results from the Calculus 86
  • 2 Functions and Identities
  • 2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 101
  • 2.2 Logarithms and Exponentials 112
  • 2.3 The Exponential Function 114
  • 2.4 Trigonometric Identities 115
  • 2.5 Hyperbolic Identities 121
  • 2.6 The Logarithm 126
  • 2.7 Inverse Trigonometric and Hyperbolic Functions 128
  • 2.8 Series Representations of Trigonometric and Hyperbolic Functions 133
  • 2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 136
  • 3 Derivatives of Elementary Functions
  • 3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 139
  • 3.2 Derivatives of Trigonometric Functions 140
  • 3.3 Derivatives of Inverse Trigonometric Functions 140
  • 3.4 Derivatives of Hyperbolic Functions 141
  • 3.5 Derivatives of Inverse Hyperbolic Functions 142
  • 4 Indefinite Integrals of Algebraic Functions
  • 4.1 Algebraic and Transcendental Functions 145
  • 4.2 Indefinite Integrals of Rational Functions 146
  • 4.3 Nonrational Algebraic Functions 158
  • 5 Indefinite Integrals of Exponential Functions
  • 5.1 Basic Results 167
  • 6 Indefinite Integrals of Logarithmic Functions
  • 6.1 Combinations of Logarithms and Polynomials 173
  • 7 Indefinite Integrals of Hyperbolic Functions
  • 7.1 Basic Results 179
  • 7.2 Integrands Involving Powers of sinh(bx) or cosh(bx) 180
  • 7.3 Integrands Involving (a [plus or minus] bx)[superscript m] sinh(cx) or (a + bx)[superscript m] cosh(cx) 181
  • 7.4 Integrands Involving x[superscript m] sinh[superscript n] x or x[superscript m] cosh[superscript n] x 183
  • 7.5 Integrands Involving x[superscript m] sinh[superscript -n] x or x[superscript m] cosh[superscript -n] x 183
  • 7.6 Integrands Involving (1 [plus or minus] cosh x)[superscript -m] 185
  • 7.7 Integrands Involving sinh(ax)cosh[superscript -n] x or cosh(ax)sinh[superscript -n] x 185
  • 7.8 Integrands Involving sinh(ax + b) and cosh(cx + d) 186
  • 7.9 Integrands Involving tanh kx and coth kx 188
  • 7.10 Integrands Involving (a + bx)[superscript m] sinh kx or (a + bx)[superscript m] cosh kx 189
  • 8 Indefinite Integrals Involving Inverse Hyperbolic Functions
  • 8.1 Basic Results 191
  • 8.2 Integrands Involving x[superscript -n] arcsinh(x/a) or x[superscript -n] arccosh(x/a) 193
  • 8.3 Integrands Involving x[superscript n] arctanh(x/a) or x[superscript n] arccoth(x/a) 194
  • 8.4 Integrands Involving x[superscript -n] arctanh(x/a) or x[superscript -n] arccoth(x/a) 195
  • 9 Indefinite Integrals of Trigonometric Functions
  • 9.1 Basic Results 197
  • 9.2 Integrands Involving Powers of x and Powers of sin x or cos x 197
  • 9.3 Integrands Involving tan x and/or cot x 205
  • 9.4 Integrands Involving sin x and cos x 207
  • 9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 211
  • 10 Indefinite Integrals of Inverse Trigonometric Functions
  • 10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 215
  • 11 The Gamma, Beta, Pi, and Psi Functions
  • 11.1 The Euler Integral and Limit and Infinite Product Representations for [Gamma] (x) 221
  • 12 Elliptic Integrals and Functions
  • 12.1 Elliptic Integrals 229
  • 12.2 Jacobian Elliptic Functions 235
  • 12.3 Derivatives and Integrals 237
  • 12.4 Inverse Jacobian Elliptic Functions 237
  • 13 Probability Integrals and the Error Function
  • 13.1 Normal Distribution 239
  • 13.2 The Error Function 242
  • 14 Fresnel Integrals, Sine and Cosine Integrals
  • 14.1 Definitions, Series Representations, and Values at Intinity 245
  • 14.2 Definitions, Series Representations, and Values at Infinity 247
  • 15 Definite Integrals
  • 15.1 Integrands Involving Powers of x 249
  • 15.2 Integrands Involving Trigonometric Functions 251
  • 15.3 Integrands Involving the Exponential Function 254
  • 15.4 Integrands Involving the Hyperbolic Function 256
  • 15.5 Integrands Involving the Logarithmic Function 256
  • 16 Different Forms of Fourier Series
  • 16.1 Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] 257
  • 16.2 Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L 258
  • 16.3 Fourier Series for f(x) on a [less than or equal] x [less than or equal] b 258
  • 16.4 Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 259
  • 16.5 Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] L 259
  • 16.6 Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 260
  • 16.7 Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] L 260
  • 16.8 Complex (Exponential) Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] 260
  • 16.9 Complex (Exponential) Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L 261
  • 16.10 Representative Examples of Fourier Series 261
  • 16.11 Fourier Series and Discontinuous Functions 265
  • 17 Bessel Functions
  • 17.1 Bessel's Differential Equation 269
  • 17.2 Series Expansions for J[subscript v](x) and Y[subscript v](x) 270
  • 17.3 Bessel Functions of Fractional Order 272
  • 17.4 Asymptotic Representations for Bessel Functions 273
  • 17.5 Zeros of Bessel Functions 273
  • 17.6 Bessel's Modified Equation 274
  • 17.7 Series Expansions for I[subscript v](x) and K[subscript v](x) 276
  • 17.8 Modified Bessel Functions of Fractional Order 277
  • 17.9 Asymptotic Representations of Modified Bessel Functions 278
  • 17.10 Relationships between Bessel Functions 278
  • 17.11 Integral Representations of J[subscript n](x), I[subscript n](x), and K[subscript n](x) 281
  • 17.12 Indefinite Integrals of Bessel Functions 281
  • 17.13 Definite Integrals Involving Bessel Functions 282
  • 17.14 Spherical Bessel Functions 283
  • 18 Orthogonal Polynomials
  • 18.2 Legendre Polynomials P[subscript n](x) 286
  • 18.3 Chebyshev Polynomials T[subscript n](x) and U[subscript n](x) 290
  • 18.4 Laguerre Polynomials L[subscript n](x) 294
  • 18.5 Hermite Polynomials H[subscript n](x) 296
  • 19 Laplace Transformation
  • 20 Fourier Transforms
  • 21 Numerical Integration
  • 21.1 Classical Methods 315
  • 22 Solutions of Standard Ordinary Differential Equations
  • 22.2 Separation of Variables 323
  • 22.3 Linear First-Order Equations 323
  • 22.4 Bernoulli's Equation 324
  • 22.5 Exact Equations 325
  • 22.6 Homogeneous Equations 325
  • 22.7 Linear Differential Equations 326
  • 22.8 Constant Coefficient Linear Differential Equations
  • Homogeneous Case 327
  • 22.9 Linear Homogeneous Second-Order Equation 330
  • 22.10 Constant Coefficient Linear Differential Equations
  • Inhomogeneous Case 331
  • 22.11 Linear Inhomogeneous Second-Order Equation 333
  • 22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients 334
  • 22.13 The Cauchy-Euler Equation 336
  • 22.14 Legendre's Equation 337
  • 22.15 Bessel's Equations 337
  • 22.16 Power Series and Frobenius Methods 339
  • 22.17 The Hypergeometric Equation 344
  • 22.18 Numerical Methods 345
  • 23 Vector Analysis
  • 23.1 Scalars and Vectors 353
  • 23.2 Scalar Products 358
  • 23.3 Vector Products 359
  • 23.4 Triple Products 360
  • 23.5 Products of Four Vectors 361
  • -- 23.6 Derivatives of Vector Functions of a Scalar t 361
  • 23.7 Derivatives of Vector Functions of Several Scalar Variables 362
  • 23.8 Integrals of Vector Functions of a Scalar Variable t 363
  • 23.9 Line Integrals 364
  • 23.10 Vector Integral Theorems 366
  • 23.11 A Vector Rate of Change Theorem 368
  • 23.12 Useful Vector Identities and Results 368
  • 24 Systems of Orthogonal Coordinates
  • 24.1 Curvilinear Coordinates 369
  • 24.2 Vector Operators in Orthogonal Coordinates 371
  • 24.3 Systems of Orthogonal Coordinates 371
  • 25 Partial Differential Equations and Special Functions
  • 25.1 Fundamental Ideas 381
  • 25.2 Method of Separation of Variables 385
  • 25.3 The Sturm-Liouville Problem and Special Functions 387
  • 25.4 A First-Order System and the Wave Equation 390
  • 25.5 Conservation Equations (Laws) 391
  • 25.6 The Method of Characteristics 392
  • 25.7 Discontinuous Solutions (Shocks) 396
  • 25.8 Similarity Solutions 398
  • 25.9 Burgers's Equation, the KdV Equation, and the KdVB Equation 400
  • 26 The z-Transform
  • 26.1 The z-Transform and Transform Pairs 403
  • 27 Numerical Approximation
  • 27.2 Economization of Series 411
  • 27.3 Pade Approximation 413
  • 27.4 Finite Difference Approximations to Ordinary and Partial Derivatives 415.

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