Handbook of Mathematical Formulas and Integrals

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. By Alan Jeffrey. Includes Bibliographical References (p. 397-399) And Index.