Jeffrey Handbook of Mathematical Formulas and Integrals

1. Auflage 2014
ISBN: 978-1-4832-9514-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 410 Seiten, Web PDF

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

Autoren/Hrsg.

Jeffrey, Alan

Inhaltsverzeichnis

1;Front Cover;1
2;Handbook of Mathematical Formulas and Integrals;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;20
6;Index of Special Functions and Notations;22
7;Chapter O. Quick Reference List of Frequently Used Data;26
7.1;0.1 Useful Identities;26
7.2;0.2 Complex Relationships;27
7.3;0.3 Constants;27
7.4;0.4 Derivatives of Elementary Functions;28
7.5;0.5 Rules of Differentiation and Integration;28
7.6;0.6 Standard Integrals;29
7.7;0.7 Standard Series;36
7.8;0.8 Geometry;38
8;Chapter 1. Numerical, Algebraic, and Analytical Results for Series and Calculus;50
8.1;1.1 Algebraic Results Involving Real and Complex Numbers;50
8.2;1.2 Finite Sums;54
8.3;1.3 Bernoulli and Euler Numbers and Polynomials;61
8.4;1.4 Determinants;70
8.5;1.5 Matrices;77
8.6;1.6 Permutations and Combinations;83
8.7;1.7 Partial Fraction Decomposition;85
8.8;1.8 Convergence of Series;88
8.9;1.9 Infinite Products;92
8.10;1.10 Functional Series;94
8.11;1.11 Power Series;96
8.12;1.12 Taylor Series;100
8.13;1.13 Fourier Series;102
8.14;1.14 Asymptotic Expansions;105
8.15;1.15 Basic Results from the Calculus;107
9;Chapter 2. Functions and Identities;122
9.1;2.1 Complex Numbers and Trigonometric and Hyperbolic Functions;122
9.2;2.2 Logarithms and Exponentials;133
9.3;2.3 The Exponential Function;134
9.4;2.4 Trigonometric Identities;136
9.5;2.5 Hyperbolic Identities;142
9.6;2.6 The Logarithm;147
9.7;2.7 Inverse Trigonometric and Hyperbolic Functions;149
9.8;2.8 Series Representations of Trigonometric and Hyperbolic Functions;154
9.9;2.9 Useful Limiting Values and Inequalities Involving Elementary Functions;157
10;Chapter 3. Derivatives of Elementary Functions;160
10.1;3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions;160
10.2;3.2 Derivatives of Trigonometric Functions;161
10.3;3.3 Derivatives of Inverse Trigonometric Functions;161
10.4;3.4 Derivatives of Hyperbolic Functions;162
10.5;3.5 Derivatives of Inverse Hyperbolic Functions;163
11;Chapter 4. Indefinite Integrals of Algebraic Functions;166
11.1;4.1 Algebraic and Transcendental Functions;166
11.2;4.2 Indefinite Integrals of Rational Functions;167
11.3;4.3 Nonrational Algebraic Functions;179
12;Chapter 5. Indefinite Integrals of Exponential Functions;188
12.1;5.1 Basic Results;188
13;Chapter 6. Indefinite Integrals of Logarithmic Functions;194
13.1;6.1 Combinations of Logarithms and Polynomials;194
14;Chapter 7. Indefinite Integrals of Hyperbolic Functions;200
14.1;7.1 Basic Results;200
14.2;7.2 Integrands Involving Powers of sinh(bx) or cosh(bx);201
14.3;7.3 Integrands Involving (a + bx)m sinh(cx) or (a + bx)m cosh(cx);202
14.4;7.4 Integrands Involving xm sinhn x or xm coshn x;204
14.5;7.5 Integrands Involving xm sinh–n x or xm cosh–n x ;204
14.6;7.6 Integrands Involving (1 ± cosh x)–m;206
14.7;7.7 Integrands Involving sinh(ax) cosh–n x or cosh(ax) sinh–n x;206
14.8;7.8 Integrands Involving sinh(ax + b) and cosh(cx + d);207
14.9;7.9 Integrands Involving tanh kx and coth kx;209
14.10;7.10 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx;210
15;Chapter 8. Indefinite Integrals Involving Inverse Hyperbolic Functions ;212
15.1;8.1 Basic Results;212
15.2;8.2 Integrands Involving x–n arcsinh(x/a) or x–n arccosh(x/a);214
15.3;8.3 Integrands Involving xn arctanh(x/a) or xn arccoth(x/a);215
15.4;8.4 Integrands Involving x–n arctanh(x/a) or x–n arccoth(x/a);216
16;Chapter 9. Indefinite Integrals of Trigonometric Functions;218
16.1;9.1 Basic Results;218
16.2;9.2 Integrands Involving Powers of x and Powers of sin x or cos x;220
16.3;9.3 Integrands Involving tan x and/or cot x;226
16.4;9.4 Integrands Involving sin x and cos x;228
16.5;9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x
;232
17;Chapter 10. Indefinite Integrals of Inverse Trigonometric Functions;236
17.1;10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions;236
18;Chapter 11. The Gamma, Beta, Pi, and Psi Functions;242
18.1;11.1 The Euler Integral and Limit and Infinite Product Representations for .(x)
;242
19;Chapter 12. Elliptic Integrals and Functions;250
19.1;12.1 Elliptic Integrals;250
19.2;12.2 Jacobian Elliptic Functions;256
19.3;12.3 Derivatives and Integrals;258
19.4;12.4 Inverse Jacobian Elliptic Functions;258
20;Chapter 13. Probability Integrals and the Error Function;260
20.1;13.1 Normal Distribution;260
20.2;13.2 The Error Function;263
21;Chapter 14. Fresnel Integrals;266
21.1;14.1 Definitions, Series Representations, and Values at Infinity;266
22;Chapter 15. Definite Integrals;268
22.1;15.1 Integrands Involving Powers of x;268
22.2;15.2 Integrands Involving Trigonometric Functions;270
22.3;15.3 Integrands Involving the Exponential Function;273
22.4;15.4 Integrands Involving the Hyperbolic Function;275
22.5;15.5 Integrands Involving the Logarithmic Function;275
23;Chapter 16. Different Forms of Fourier Series;278
23.1;16.1 Fourier Series for f(x) on –p = x = p
;278
23.2;16.2 Fourier Series for f(x) on –L = x = L
;279
23.3;16.3 Fourier Series for f(x) on a = x = b
;279
23.4;16.4 Half-Range Fourier Cosine Series for f(x) on 0 = x = L
;280
23.5;16.5 Half-Range Fourier Cosine Series for f(x) on 0 = x = L
;280
23.6;16.6 Half-Range Fourier Sine Series for f(x) on 0 = x = L
;281
23.7;16.7 Half-Range Fourier Sine Series for f(x) on 0 = x = L
;281
23.8;16.8 Complex (Exponential) Fourier Series for f(x) on –p = x = p
;281
23.9;16.9 Complex (Exponential) Fourier Series for f(x) on — L = x = L
;282
23.10;16.10 Representative Examples of Fourier Series;282
23.11;16.11 Fourier Series and Discontinuous Functions;286
24;Chapter 17. Bessel Functions;290
24.1;17.1 Bessel's Differential Equation;290
24.2;17.2 Series Expansions for Jv(x) and Yv(x);291
24.3;17.3 Bessel Functions of Fractional Order;293
24.4;17.4 Asymptotic Representations of Bessel Functions;294
24.5;17.5 Zeros of Bessel Functions;294
24.6;17.6 Bessel's Modified Equation;296
24.7;17.7 Series Expansions for Iv(x) and Kv(x);297
24.8;17.8 Modified Bessel Functions of Fractional Order;298
24.9;17.9 Asymptotic Representations of Modified Bessel Functions;299
24.10;17.10 Relationships between Bessel Functions;300
24.11;17.11 Integral Representations of Jn(x), In (x), and Kn (x);302
24.12;17.12 Indefinite Integrals of Bessel Functions;303
24.13;17.13 Definite Integrals Involving Bessel Functions;304
25;Chapter 18. Orthogonal Polynomials;306
25.1;18.1 Introduction;306
25.2;18.2 Legendre Polynomials Pn(x);307
25.3;18.3 Chebyshev Polynomials Tn(x) and Un(x);311
25.4;18.4 Laguerre Polynomials Ln(x);314
25.5;18.5 Hermite Polynomials Hn(x);316
26;Chapter 19. Laplace Transformation;318
26.1;19.1 Introduction;318
27;Chapter 20. Fourier Transforms;326
27.1;20.1 Introduction;326
28;Chapter 21. Numerical Integration;334
28.1;21.1 Classical Methods;334
29;Chapter 22. Solutions of Standard Ordinary Differential Equations;340
29.1;22.1 Introduction;340
29.2;22.2 Separation of Variables;342
29.3;22.3 Linear First-Order Equations;342
29.4;22.4 Bernoulli's Equation;343
29.5;22.5 Exact Equations;344
29.6;22.6 Homogeneous Equations;344
29.7;22.7 Linear Differential Equations;345
29.8;22.8 Constant Coefficient Linear Differential Equations—Homogeneous Case;346
29.9;22.9 Linear Homogeneous Second-Order Equation;349
29.10;22.10 Constant Coefficient Linear Differential Equations—Inhomogeneous Case;350
29.11;22.11 Linear Inhomogeneous Second-Order Equation;352
29.12;22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients;353
29.13;22.13 The Cauchy–Euler Equation;355
29.14;22.14 Legendre's Equation;356
29.15;22.15 Bessel's Equations;356
29.16;22.16 Power Series and Frobenius Methods;358
29.17;22.17 The Hypergeometric Equation;363
29.18;22.18 Numerical Methods;364
30;Chapter 23. Vector Analysis;372
30.1;23.1 Scalars and Vectors;372
30.2;23.2 Scalar Products;377
30.3;23.3 Vector Products;378
30.4;23.4 Triple Products;379
30.5;23.5 Products of Four Vectors;380
30.6;23.6 Derivatives of Vector Functions of a Scalar t;380
30.7;23.7 Derivatives of Vector Functions of Several Scalar Variables;381
30.8;23.8 Integrals of Vector Functions of a Scalar Variable t;382
30.9;23.9 Line Integrals;383
30.10;23.10 Vector Integral Theorems;385
30.11;23.11 A Vector Rate of Change Theorem;387
30.12;23.12 Useful Vector Identities and Results;387
31;Chapter 24. Systems of Orthogonal Coordinates
;388
31.1;24.1 Curvilinear Coordinates;388
31.2;24.2 Vector Operators in Orthogonal Coordinates;390
31.3;24.3 Systems of Orthogonal Coordinates;390
32;Chapter 25. Partial Differential Equations and Special Functions;400
32.1;25.1 Fundamental Ideas;400
32.2;25.2 Method of Separation of Variables;404
32.3;25.3 The Sturm–Liouville Problem and Special Functions
;406
32.4;25.4 A First-Order System and the Wave Equation;409
32.5;25.5 Conservation Equations (Laws);410
32.6;25.6 The Method of Characteristics;411
32.7;25.7 Discontinuous Solutions (Shocks);415
32.8;25.8 Similarity Solutions;417
32.9;25.9 Burgers's Equation, the KdV Equation, and the KdVB Equation;419
33;Short Classified Reference List;422
34;Index;426