Handbook of Mathematical Formulas and Integrals (eBook)

Front Cover 1
MATHEMATICAL FORMULAS AND INTEGRALS 4
Copyright Page 5
Contents 6
Preface 20
Preface to the Second Edition 22
Index of Special Functions and Notations 24
Chapter 0. Quick Reference List of Frequently Used Data 28
0.1 Useful Identities 28
0.2 Complex Relationships 29
0.3 Constants 29
0.4 Derivatives of Elementary Functions 30
0.5 Rules of Differentiation and Integration 30
0.6 Standard Integrals 31
0.7 Standard Series 38
0.8 Geometry 40
Chapter 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 52
1.1 Algebraic Results Involving Real and Complex Numbers 52
1.2 Finite Sums 56
1.3 Bernoulli and Euler Numbers and Polynomials 64
1.4 Determinants 74
1.5 Matrices 82
1.6 Permutations and Combinations 89
1.7 Partial Fraction Decomposition 90
1.8 Convergence of Series 93
1.9 Infinite Products 98
1.10 Functional Series 100
1.11 Power Series 101
1.12 Taylor Series 106
1.13 Fourier Series 108
1.14 Asymptotic Expansions 112
1.15 Basic Results from the Calculus 113
Chapter 2. Functions and Identities 128
2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 128
2.2 Logarithms and Exponentials 139
2.3 The Exponential Function 141
2.4 Trigonometric Identities 142
2.5 Hyperbolic Identities 148
2.6 The Logarithm 153
2.7 Inverse Trigonometric and Hyperbolic Functions 155
2.8 Series Representations of Trigonometric and Hyperbolic Functions 160
2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 163
Chapter 3. Derivatives of Elementary Functions 166
3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 166
3.2 Derivatives of Trigonometric Functions 167
3.3 Derivatives of Inverse Trigonometric Functions 167
3.4 Derivatives of Hyperbolic Functions 168
3.5 Derivatives of Inverse Hyperbolic Functions 169
Chapter 4. Indefinite Integrals of Algebraic Functions 172
4.1 Algebraic and Transcendental Functions 172
4.2 Indefinite Integrals of Rational Functions 173
4.3 Nonrational Algebraic Functions 185
Chapter 5 Indefinite Integrals of Exponential Functions 194
5.1 Basic Results 194
Chapter 6. Indefinite Integrals of Logarithmic Functions 200
6.1 Combinations of Logarithms and Polynomials 200
Chapter 7. Indefinite Integrals of Hyperbolic Functions 206
7.1 Basic Results 206
7.2 Integrands Involving Powers of sinh(bx) or cosh(bx) 207
7.3 Integrands Involving (a ± bx)m sinh(cx) or (a + bx)m cosh(cx) 208
7.4 Integrands Involving xm sinhnx or xm coshnx 210
7.5 Integrands Involving xm sinh-nx or xm cosh-nx 210
7.6 Integrands Involving (1 ± cosh x)-m 212
7.7 Integrands Involving sinh(ax)cosh-nx or cosh(ax)sinh-nx 212
7.8 Integrands Involving sinh(ax + b) and cosh(cx + d) 213
7.9 Integrands Involving tanh kx and coth kx 215
7.10 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 216
Chapter 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 218
8.1 Basic Results 218
8.2 Integrands Involving x-n arcsinh(x/a) or x-n arccosh(x/a) 220
8.3 Integrands Involving xn arctanh(x/a) or xn arccoth(x/a) 221
8.4 Integrands Involving x-n arctanh(x/a) or x-n arccoth(x/a) 222
Chapter 9. Indefinite Integrals of Trigonometric Functions 224
9.1 Basic Results 224
9.2 Integrands Involving Powers of x and Powers of sin x or cos x 226
9.3 Integrands Involving tan x and/or cot x 232
9.4 Integrands Involving sin x and cos x 234
9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 211 238
Chapter 10. Indefinite Integrals of Inverse Trigonometric Functions 242
10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 242
Chapter 11. The Gamma, Beta, Pi, and Psi Functions 248
11.1 The Euler Integral and Limit and Infinite Product Representations for (x) 248
Chapter 12. Elliptic Integrals and Functions 256
12.1 Elliptic Integrals 256
12.2 Jacobian Elliptic Functions 262
12.3 Derivatives and Integrals 264
12.4 Inverse Jacobian Elliptic Functions 264
Chapter 13. Probability Integrals and the Error Function 266
13.1 Normal Distribution 266
13.2 The Error Function 269
Chapter 14. Fresnel Integrals, Sine and Cosine Integrals 272
14.1 Definitions, Series Representations, and Values at Infinity 272
14.2 Definitions, Series Representations, and Values at Innity 274
Chapter 15. Definite Integrals 276
15.1 Integrands Involving Powers of x 276
15.2 Integrands Involving Trigonometric Functions 278
15.3 Integrands Involving the Exponential Function 281
15.4 Integrands Involving the Hyperbolic Function 283
15.5 Integrands Involving the Logarithmic Function 283
Chapter 16. Different Forms of Fourier Series 284
16.1 Fourier Series for f (x) on p = x = p 284
16.2 Fourier Series for f (x) on L = x = L 285
16.3 Fourier Series for f (x) on a = x = b 285
16.4 Half-Range Fourier Cosine Series for f (x) on 0 = x = p 286
16.5 Half-Range Fourier Cosine Series for f (x) on 0 = x = L 286
16.6 Half-Range Fourier Sine Series for f (x) on 0 = x = p 287
16.7 Half-Range Fourier Sine Series for f (x) on 0 = x = L 287
16.8 Complex (Exponential) Fourier Series for f (x) on p = x = p 287
16.9 Complex (Exponential) Fourier Series for f (x) on L = x = L 288
16.10 Representative Examples of Fourier Series 288
16.11 Fourier Series and Discontinuous Functions 292
Chapter 17. Bessel Functions 296
17.1 Bessel’s Differential Equation 296
17.2 Series Expansions for J.(x) and Y.(x) 297
17.3 Bessel Functions of Fractional Order 299
17.4 Asymptotic Representations for Bessel Functions 300
17.5 Zeros of Bessel Functions 300
17.6 Bessel’s Modified Equation 301
17.7 Series Expansions for I.(x) and K.(x) 303
17.8 Modified Bessel Functions of Fractional Order 304
17.9 Asymptotic Representations of Modified Bessel Functions 305
17.10 Relationships between Bessel Functions 305
17.11 Integral Representations of Jn(x), In(x), and Kn(x) 308
17.12 Indefinite Integrals of Bessel Functions 308
17.13 Definite Integrals Involving Bessel Functions 309
17.14 Spherical Bessel Functions 310
Chapter 18. Orthogonal Polynomials 312
18.1 Introduction 312
18.2 Legendre Polynomials Pn(x) 313
18.3 Chebyshev Polynomials Tn(x) and Un(x) 317
18.4 Laguerre Polynomials Ln(x) 321
18.5 Hermite Polynomials Hn(x) 323
Chapter 19. Laplace Transformation 326
19.1 Introduction 326
Chapter 20. Fourier Transforms 334
20.1 Introduction 334
Chapter 21. Numerical Integration 342
21.1 Classical Methods 342
Chapter 22. Solutions of Standard Ordinary Differential Equations 348
22.1 Introduction 348
22.2 Separation of Variables 350
22.3 Linear First-Order Equations 350
22.4 Bernoulli’s Equation 351
22.5 Exact Equations 352
22.6 Homogeneous Equations 352
22.7 Linear Differential Equations 353
22.8 Constant Coefficient Linear Differential Equations—Homogeneous Case 354
22.9 Linear Homogeneous Second-Order Equation 357
22.10 Constant Coefficient Linear Differential Equations—Inhomogeneous Case 358
22.11 Linear Inhomogeneous Second-Order Equation 360
22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients 361
22.13 The Cauchy–Euler Equation 363
22.14 Legendre’s Equation 364
22.15 Bessel’s Equations 364
22.16 Power Series and Frobenius Methods 366
22.17 The Hypergeometric Equation 371
22.18 Numerical Methods 372
Chapter 23. Vector Analysis 380
23.1 Scalars and Vectors 380
23.2 Scalar Products 385
23.3 Vector Products 386
23.4 Triple Products 387
23.5 Products of Four Vectors 388
23.6 Derivatives of Vector Functions of a Scalar t 388
23.7 Derivatives of Vector Functions of Several Scalar Variables 389
23.8 Integrals of Vector Functions of a Scalar Variable t 390
23.9 Line Integrals 391
23.10 Vector Integral Theorems 393
23.11 A Vector Rate of Change Theorem 395
23.12 Useful Vector Identities and Results 395
Chapter 24 Systems of Orthogonal Coordinates 396
24.1 Curvilinear Coordinates 396
24.2 Vector Operators in Orthogonal Coordinates 398
24.3 Systems of Orthogonal Coordinates 398
Chapter 25. Partial Differential Equations and Special Functions 408
25.1 Fundamental Ideas 408
25.2 Method of Separation of Variables 412
25.3 The Sturm–Liouville Problem and Special Functions 414
25.4 A First-Order System and the Wave Equation 417
25.5 Conservation Equations (Laws) 418
25.6 The Method of Characteristics 419
25.7 Discontinuous Solutions (Shocks) 423
25.8 Similarity Solutions 425
25.9 Burgers’s Equation, the KdV Equation, and the KdVB Equation 427
Chapter 26. The z-Transform 430
26.1 The z -Transform and Transform Pairs 430
Chapter 27. Numerical Approximation 436
27.1 Introduction 436
27.2 Economization of Series 438
27.3 Padé Approximation 440
27.4 Finite Difference Approximations to Ordinary and Partial Derivatives 442
Short Classified Reference List 446
Index 450