Handbook of Mathematical Formulas and Integrals (eBook)

Alan Jeffrey (Autor)

eBook Download: PDF

2014 | 1. Auflage
410 Seiten
Elsevier Science (Verlag)
978-1-4832-9514-5 (ISBN)

Infos

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 formulas, functions, relations, and methods from algebra, trigonometric and exponential functions, combinatorics, probability, matrix theory, calculus and vector calculus, both ordinary and partial differential equations, Fourier series, orthogonal polynomials, and Laplace transforms. Based on Gradshteyn and Ryzhik's Table of Integrals, Series, and Products, Fifth Edition (edited by Jeffrey), but far more accessible and written with particular attention to the needs of students and practicing scientists and engineers, this book is an essential resource. Affordable and authoritative, it is the first place to look for help and a rewarding place to browse.Special thumb-tab index throughout the book for ease of useAnswers are keyed to the type of problem they solveFormulas are provided for problems across the entire spectrum of MathematicsAll equations are sent from a computer-checked source codeCompanion to Gradshteyn: Table of Integrals, Series, and Products, Fifth EditionThe following features make the Handbook a Better Value than its Competition:Less expensiveMore comprehensiveEquations are computer-validated with Scientific WorkPlace(tm) and Mathematica(r)Superior quality from one of the most respected names in scientific and technical publishingOffers unique thumb-tab indexing throughout the book which makes finding answers quick and easy

Inhalt

Front Cover 1
Handbook of Mathematical Formulas and Integrals 4
Copyright Page 5
Table of Contents 6
Preface 20
Index of Special Functions and Notations 22
Chapter O. Quick Reference List of Frequently Used Data 26
0.1 Useful Identities 26
0.2 Complex Relationships 27
0.3 Constants 27
0.4 Derivatives of Elementary Functions 28
0.5 Rules of Differentiation and Integration 28
0.6 Standard Integrals 29
0.7 Standard Series 36
0.8 Geometry 38
Chapter 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 50
1.1 Algebraic Results Involving Real and Complex Numbers 50
1.2 Finite Sums 54
1.3 Bernoulli and Euler Numbers and Polynomials 61
1.4 Determinants 70
1.5 Matrices 77
1.6 Permutations and Combinations 83
1.7 Partial Fraction Decomposition 85
1.8 Convergence of Series 88
1.9 Infinite Products 92
1.10 Functional Series 94
1.11 Power Series 96
1.12 Taylor Series 100
1.13 Fourier Series 102
1.14 Asymptotic Expansions 105
1.15 Basic Results from the Calculus 107
Chapter 2. Functions and Identities 122
2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 122
2.2 Logarithms and Exponentials 133
2.3 The Exponential Function 134
2.4 Trigonometric Identities 136
2.5 Hyperbolic Identities 142
2.6 The Logarithm 147
2.7 Inverse Trigonometric and Hyperbolic Functions 149
2.8 Series Representations of Trigonometric and Hyperbolic Functions 154
2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 157
Chapter 3. Derivatives of Elementary Functions 160
3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 160
3.2 Derivatives of Trigonometric Functions 161
3.3 Derivatives of Inverse Trigonometric Functions 161
3.4 Derivatives of Hyperbolic Functions 162
3.5 Derivatives of Inverse Hyperbolic Functions 163
Chapter 4. Indefinite Integrals of Algebraic Functions 166
4.1 Algebraic and Transcendental Functions 166
4.2 Indefinite Integrals of Rational Functions 167
4.3 Nonrational Algebraic Functions 179
Chapter 5. Indefinite Integrals of Exponential Functions 188
5.1 Basic Results 188
Chapter 6. Indefinite Integrals of Logarithmic Functions 194
6.1 Combinations of Logarithms and Polynomials 194
Chapter 7. Indefinite Integrals of Hyperbolic Functions 200
7.1 Basic Results 200
7.2 Integrands Involving Powers of sinh(bx) or cosh(bx) 201
7.3 Integrands Involving (a + bx)m sinh(cx) or (a + bx)m cosh(cx) 202
7.4 Integrands Involving xm sinhn x or xm coshn x 204
7.5 Integrands Involving xm sinh–n x or xm cosh–n x 204
7.6 Integrands Involving (1 ± cosh x)–m 206
7.7 Integrands Involving sinh(ax) cosh–n x or cosh(ax) sinh–n x 206
7.8 Integrands Involving sinh(ax + b) and cosh(cx + d) 207
7.9 Integrands Involving tanh kx and coth kx 209
7.10 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 210
Chapter 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 212
8.1 Basic Results 212
8.2 Integrands Involving x–n arcsinh(x/a) or x–n arccosh(x/a) 214
8.3 Integrands Involving xn arctanh(x/a) or xn arccoth(x/a) 215
8.4 Integrands Involving x–n arctanh(x/a) or x–n arccoth(x/a) 216
Chapter 9. Indefinite Integrals of Trigonometric Functions 218
9.1 Basic Results 218
9.2 Integrands Involving Powers of x and Powers of sin x or cos x 220
9.3 Integrands Involving tan x and/or cot x 226
9.4 Integrands Involving sin x and cos x 228
9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 
232 
Chapter 10. Indefinite Integrals of Inverse Trigonometric Functions 236
10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 236
Chapter 11. The Gamma, Beta, Pi, and Psi Functions 242
11.1 The Euler Integral and Limit and Infinite Product Representations for .(x) 
242 
Chapter 12. Elliptic Integrals and Functions 250
12.1 Elliptic Integrals 250
12.2 Jacobian Elliptic Functions 256
12.3 Derivatives and Integrals 258
12.4 Inverse Jacobian Elliptic Functions 258
Chapter 13. Probability Integrals and the Error Function 260
13.1 Normal Distribution 260
13.2 The Error Function 263
Chapter 14. Fresnel Integrals 266
14.1 Definitions, Series Representations, and Values at Infinity 266
Chapter 15. Definite Integrals 268
15.1 Integrands Involving Powers of x 268
15.2 Integrands Involving Trigonometric Functions 270
15.3 Integrands Involving the Exponential Function 273
15.4 Integrands Involving the Hyperbolic Function 275
15.5 Integrands Involving the Logarithmic Function 275
Chapter 16. Different Forms of Fourier Series 278
16.1 Fourier Series for f(x) on –p = x = p 
278 
16.2 Fourier Series for f(x) on –L = x = L 
279 
16.3 Fourier Series for f(x) on a = x = b 
279 
16.4 Half-Range Fourier Cosine Series for f(x) on 0 = x = L 
280 
16.5 Half-Range Fourier Cosine Series for f(x) on 0 = x = L 
280 
16.6 Half-Range Fourier Sine Series for f(x) on 0 = x = L 
281 
16.7 Half-Range Fourier Sine Series for f(x) on 0 = x = L 
281 
16.8 Complex (Exponential) Fourier Series for f(x) on –p = x = p 
281 
16.9 Complex (Exponential) Fourier Series for f(x) on — L = x = L 
282 
16.10 Representative Examples of Fourier Series 282
16.11 Fourier Series and Discontinuous Functions 286
Chapter 17. Bessel Functions 290
17.1 Bessel's Differential Equation 290
17.2 Series Expansions for Jv(x) and Yv(x) 291
17.3 Bessel Functions of Fractional Order 293
17.4 Asymptotic Representations of Bessel Functions 294
17.5 Zeros of Bessel Functions 294
17.6 Bessel's Modified Equation 296
17.7 Series Expansions for Iv(x) and Kv(x) 297
17.8 Modified Bessel Functions of Fractional Order 298
17.9 Asymptotic Representations of Modified Bessel Functions 299
17.10 Relationships between Bessel Functions 300
17.11 Integral Representations of Jn(x), In (x), and Kn (x) 302
17.12 Indefinite Integrals of Bessel Functions 303
17.13 Definite Integrals Involving Bessel Functions 304
Chapter 18. Orthogonal Polynomials 306
18.1 Introduction 306
18.2 Legendre Polynomials Pn(x) 307
18.3 Chebyshev Polynomials Tn(x) and Un(x) 311
18.4 Laguerre Polynomials Ln(x) 314
18.5 Hermite Polynomials Hn(x) 316
Chapter 19. Laplace Transformation 318
19.1 Introduction 318
Chapter 20. Fourier Transforms 326
20.1 Introduction 326
Chapter 21. Numerical Integration 334
21.1 Classical Methods 334
Chapter 22. Solutions of Standard Ordinary Differential Equations 340
22.1 Introduction 340
22.2 Separation of Variables 342
22.3 Linear First-Order Equations 342
22.4 Bernoulli's Equation 343
22.5 Exact Equations 344
22.6 Homogeneous Equations 344
22.7 Linear Differential Equations 345
22.8 Constant Coefficient Linear Differential Equations—Homogeneous Case 346
22.9 Linear Homogeneous Second-Order Equation 349
22.10 Constant Coefficient Linear Differential Equations—Inhomogeneous Case 350
22.11 Linear Inhomogeneous Second-Order Equation 352
22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients 353
22.13 The Cauchy–Euler Equation 355
22.14 Legendre's Equation 356
22.15 Bessel's Equations 356
22.16 Power Series and Frobenius Methods 358
22.17 The Hypergeometric Equation 363
22.18 Numerical Methods 364
Chapter 23. Vector Analysis 372
23.1 Scalars and Vectors 372
23.2 Scalar Products 377
23.3 Vector Products 378
23.4 Triple Products 379
23.5 Products of Four Vectors 380
23.6 Derivatives of Vector Functions of a Scalar t 380
23.7 Derivatives of Vector Functions of Several Scalar Variables 381
23.8 Integrals of Vector Functions of a Scalar Variable t 382
23.9 Line Integrals 383
23.10 Vector Integral Theorems 385
23.11 A Vector Rate of Change Theorem 387
23.12 Useful Vector Identities and Results 387
Chapter 24. Systems of Orthogonal Coordinates 
388 
24.1 Curvilinear Coordinates 388
24.2 Vector Operators in Orthogonal Coordinates 390
24.3 Systems of Orthogonal Coordinates 390
Chapter 25. Partial Differential Equations and Special Functions 400
25.1 Fundamental Ideas 400
25.2 Method of Separation of Variables 404
25.3 The Sturm–Liouville Problem and Special Functions 
406 
25.4 A First-Order System and the Wave Equation 409
25.5 Conservation Equations (Laws) 410
25.6 The Method of Characteristics 411
25.7 Discontinuous Solutions (Shocks) 415
25.8 Similarity Solutions 417
25.9 Burgers's Equation, the KdV Equation, and the KdVB Equation 419
Short Classified Reference List 422
Index 426

Produkt-Details

• Erscheint lt. Verlag: 19.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-9514-1 / 1483295141
• ISBN-13: 978-1-4832-9514-5 / 9781483295145