Handbook of Mathematical Formulas and Integrals (eBook)

Front Cover 1
Handbook of Mathematical Formulas and Integrals 4
Copyright Page 5
Table of Contents 6
Preface 20
Preface to the Fourth Edition 22
Notes for Handbook Users 24
Index of Special Functions and Notations 44
Chapter 0. Quick Reference List of Frequently Used Data 48
0.1 Useful Identities 48
0.2 Complex Relationships 49
0.3 Constants, Binomial Coefficients and the Pochhammer Symbol 50
0.4 Derivatives of Elementary Functions 50
0.5 Rules of Differentiation and Integration 51
0.6 Standard Integrals 51
0.7 Standard Series 57
0.8 Geometry 59
Chapter 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 74
1.1 Algebraic Results Involving Real and Complex Numbers 74
1.2 Finite Sums 79
1.3 Bernoulli and Euler Numbers and Polynomials 87
1.4 Determinants 97
1.5 Matrices 105
1.6 Permutations and Combinations 114
1.7 Partial Fraction Decomposition 115
1.8 Convergence of Series 119
1.9 Infinite Products 124
1.10 Functional Series 126
1.11 Power Series 129
1.12 Taylor Series 133
1.13 Fourier Series 136
1.14 Asymptotic Expansions 140
1.15 Basic Results from the Calculus 142
Chapter 2. Functions and Identities 156
2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 156
2.2 Logarithms and Exponentials 168
2.3 The Exponential Function 170
2.4 Trigonometric Identities 171
2.5 Hyperbolic Identities 179
2.6 The Logarithm 184
2.7 Inverse Trigonometric and Hyperbolic Functions 186
2.8 Series Representations of Trigonometric and Hyperbolic Functions 191
2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 194
Chapter 3. Derivatives of Elementary Functions 196
3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 196
3.2 Derivatives of Trigonometric Functions 197
3.3 Derivatives of Inverse Trigonometric Functions 197
3.4 Derivatives of Hyperbolic Functions 198
3.5 Derivatives of Inverse Hyperbolic Functions 199
Chapter 4. Indefinite Integrals of Algebraic Functions 200
4.1 Algebraic and Transcendental Functions 200
4.2 Indefinite Integrals of Rational Functions 201
4.3 Nonrational Algebraic Functions 213
Chapter 5. Indefinite Integrals of Exponential Functions 222
5.1 Basic Results 222
Chapter 6. Indefinite Integrals of Logarithmic Functions 228
6.1 Combinations of Logarithms and Polynomials 228
Chapter 7. Indefinite Integrals of Hyperbolic Functions 236
7.1 Basic Results 236
7.2 Integrands Involving Powers of sinh(bx) or cosh(bx) 237
7.3 Integrands Involving (a + bx)m sinh(cx) or (a + bx)m cosh(cx) 238
7.4 Integrands Involving xm sinhn x or xm coshn x 240
7.5 Integrands Involving xm sinh –n x or xm cosh –n x 240
7.6 Integrands Involving (1 ± cosh x) –m 242
7.7 Integrands Involving sinh(ax) cosh –n x or cosh(ax) sinh –n x 242
7.8 Integrands Involving sinh(ax + b) and cosh(cx + d) 243
7.9 Integrands Involving tanh kx and coth kx 245
7.10 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 246
Chapter 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 248
8.1 Basic Results 248
8.2 Integrands Involving x –n arcsinh (x/a) or x –n arccosh (x/a) 249
8.3 Integrands Involving xn arctanh (x/a) or xn arccoth (x/a) 251
8.4 Integrands Involving x –n arctanh (x/a) or x –n arccoth (x/a) 252
Chapter 9. Indefinite Integrals of Trigonometric Functions 254
9.1 Basic Results 254
9.2 Integrands Involving Powers of x and Powers of sin x or cos x 256
9.3 Integrands Involving tan x and/or cot x 262
9.4 Integrands Involving sin x and cos x 264
9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 268
Chapter 10. Indefinite Integrals of Inverse Trigonometric Functions 272
10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 272
Chapter 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions 278
11.1 The Euler Integral. Limit and Infinite Product Representations for the Gamma Function (x). The Incomplete Gamma Functions (a, x) and . (a, x) 278
Chapter 12. Elliptic Integrals and Functions 288
12.1 Elliptic Integrals 288
12.2 Jacobian Elliptic Functions 294
12.3 Derivatives and Integrals 296
12.4 Inverse Jacobian Elliptic Functions 297
Chapter 13. Probability Distributions and Integrals, and the Error Function 300
13.1 Distributions 300
13.2 The Error Function 304
Chapter 14. Fresnel Integrals, Sine and Cosine Integrals 308
14.1 Definitions, Series Representations, and Values at Infinity 308
14.2 Definitions, Series Representations, and Values at Infinity 310
Chapter 15. Definite Integrals 312
15.1 Integrands Involving Powers of x 312
15.2 Integrands Involving Trigonometric Functions 314
15.3 Integrands Involving the Exponential Function 317
15.4 Integrands Involving the Hyperbolic Function 320
15.5 Integrands Involving the Logarithmic Function 320
15.6 Integrands Involving the Exponential Integral Ei(x) 321
Chapter 16. Different Forms of Fourier Series 322
16.1 Fourier Series for f (x) on -p = x = p 322
16.2 Fourier Series for f (x) on -L = x = L 323
16.3 Fourier Series for f (x) on a = x = b 323
16.4 Half-Range Fourier Cosine Series for f (x) on 0 = x = p 324
16.5 Half-Range Fourier Cosine Series for f (x) on 0 = x = L 324
16.6 Half-Range Fourier Sine Series for f (x) on 0 = x = p 325
16.7 Half-Range Fourier Sine Series for f (x) on 0 = x = L 325
16.8 Complex (Exponential) Fourier Series for f (x) on -p = x = p 326
16.9 Complex (Exponential) Fourier Series for f (x) on -L = x = L 326
16.10 Representative Examples of Fourier Series 327
16.11 Fourier Series and Discontinuous Functions 332
Chapter 17. Bessel Functions 336
17.1 Bessel‘s Differential Equation 336
17.2 Series Expansions for J.(x) and Y.(x) 337
17.3 Bessel Functions of Fractional Order 339
17.4 Asymptotic Representations for Bessel Functions 341
17.5 Zeros of Bessel Functions 341
17.6 Bessel‘s Modified Equation 341
17.7 Series Expansions for I.(x) and K.(x) 344
17.8 Modified Bessel Functions of Fractional Order 345
17.9 Asymptotic Representations of Modified Bessel Functions 346
17.10 Relationships Between Bessel Functions 346
17.11 Integral Representations of Jn(x), In(x), and Kn(x) 349
17.12 Indefinite Integrals of Bessel Functions 349
17.13 Definite Integrals Involving Bessel Functions 350
17.14 Spherical Bessel Functions 351
17.15 Fourier-Bessel Expansions 354
Chapter 18. Orthogonal Polynomials 356
18.1 Introduction 356
18.2 Legendre Polynomials Pn(x) 357
18.3 Chebyshev Polynomials Tn(x) and Un(x) 367
18.4 Laguerre Polynomials Ln(x) 372
18.5 Hermite Polynomials Hn(x) 376
18.6 Jacobi Polynomials P(a,ß)n (x) 379
Chapter 19. Laplace Transformation 384
19.1 Introduction 384
Chapter 20. Fourier Transforms 400
20.1 Introduction 400
Chapter 21. Numerical Integration 410
21.1 Classical Methods 410
Chapter 22. Solutions of Standard Ordinary Differential Equations 418
22.1 Introduction 418
22.2 Separation of Variables 420
22.3 Linear First-Order Equations 420
22.4 Bernoulli‘s Equation 421
22.5 Exact Equations 422
22.6 Homogeneous Equations 423
22.7 Linear Differential Equations 423
22.8 Constant Coefficient Linear Differential Equations—Homogeneous Case 424
22.9 Linear Homogeneous Second-Order Equation 428
22.10 Linear Differential Equations—Inhomogeneous Case and the Green’s Function 429
22.11 Linear Inhomogeneous Second-Order Equation 436
22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients 437
22.13 The Cauchy–Euler Equation 440
22.14 Legendre‘s Equation 441
22.15 Bessel‘s Equations 441
22.16 Power Series and Frobenius Methods 443
22.17 The Hypergeometric Equation 450
22.18 Numerical Methods 451
Chapter 23. Vector Analysis 462
23.1 Scalars and Vectors 462
23.2 Scalar Products 467
23.3 Vector Products 468
23.4 Triple Products 469
23.5 Products of Four Vectors 470
23.6 Derivatives of Vector Functions of a Scalar t 470
23.7 Derivatives of Vector Functions of Several Scalar Variables 472
23.8 Integrals of Vector Functions of a Scalar Variable t 473
23.9 Line Integrals 474
23.10 Vector Integral Theorems 475
23.11 A Vector Rate of Change Theorem 478
23.12 Useful Vector Identities and Results 478
Chapter 24. Systems of Orthogonal Coordinates 480
24.1 Curvilinear Coordinates 480
24.2 Vector Operators in Orthogonal Coordinates 482
24.3 Systems of Orthogonal Coordinates 483
Chapter 25. Partial Differential Equations and Special Functions 494
25.1 Fundamental Ideas 494
25.2 Method of Separation of Variables 498
25.3 The Sturm–Liouville Problem and Special Functions 500
25.4 A First-Order System and the Wave Equation 503
25.5 Conservation Equations (Laws) 504
25.6 The Method of Characteristics 505
25.7 Discontinuous Solutions (Shocks) 509
25.8 Similarity Solutions 512
25.9 Burgers‘s Equation, the KdV Equation, and the KdVB Equation 514
25.10 The Poisson Integral Formulas 517
25.11 The Riemann Method 518
Chapter 26. Qualitative Properties of the Heat and Laplace Equation 520
26.1 The Weak Maximum/Minimum Principle for the Heat Equation 520
26.2 The Maximum/Minimum Principle for the Laplace Equation 520
26.3 Gauss Mean Value Theorem for Harmonic Functions in the Plane 520
26.4 Gauss Mean Value Theorem for Harmonic Functions in Space 521
Chapter 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 522
27.1 Elliptic Equations (The Laplace Equation) 522
27.2 Parabolic Equations (The Heat or Diffusion Equation) 529
27.3 Hyperbolic Equations (Wave Equation) 535
Chapter 28. The z-Transform 540
28.1 The z-Transform and Transform Pairs 540
Chapter 29. Numerical Approximation 546
29.1 Introduction 546
29.2 Economization of Series 548
29.3 Padé Approximation 550
29.4 Finite Difference Approximations to Ordinary and Partial Derivatives 552
Chapter 30. Conformal Mapping and Boundary Value Problems 556
30.1 Analytic Functions and the Cauchy-Riemann Equations 556
30.2 Harmonic Conjugates and the Laplace Equation 557
30.3 Conformal Transformations and Orthogonal Trajectories 557
30.4 Boundary Value Problems 558
30.5 Some Useful Conformal Mappings 559
Short Classified Reference List 572
Index 576