|
| Titre : |
Table of integrals, series and products |
| Type de document : |
texte imprimé |
| Auteurs : |
Gradshtein, Izrail Solomonovich, Auteur ; Iosif Moiseevich Ryzhik, Auteur |
| Mention d'édition : |
7e ed. |
| Editeur : |
Amsterdam : Elsevier |
| Année de publication : |
2007 |
| Importance : |
1 vol (1171 p.) |
| Présentation : |
ill. |
| Format : |
25 cm |
| Accompagnement : |
1 CD-ROM (sd., col. ; 4 3/4 in.) |
| ISBN/ISSN/EAN : |
978-0-12-373637-6 |
| Note générale : |
978-0-12-373637-6 |
| Langues : |
Anglais (eng) Langues originales : Anglais (eng) |
| Catégories : |
Mathématique
|
| Mots-clés : |
Intégrales : Tables
Mathématiques : Tables
Séries (mathématiques) : Tables |
| Index. décimale : |
515-Analyse mathèmatique |
| Résumé : |
Le tableau des intégrales, des séries et des produits est la référence essentielle pour les intégrales en anglais. Les mathématiciens, les scientifiques et les ingénieurs s'en remettent à l'identification et à la résolution de problèmes extrêmement complexes. Depuis la publication de la première édition en langue anglaise en 1965, elle a été entièrement révisée et agrandie régulièrement, avec des ajouts importants et, le cas échéant, des écritures existantes corrigées ou révisées. La septième édition comprend un CD-Rom entièrement consultable. |
| Note de contenu : |
Sommaire
0 Introduction 1
0.1 Finite Sums ..................................... 1
0.11 Progressions .................................... 1
0.12 Sums of powers of natural numbers ........................ 1
0.13 Sums of reciprocals of natural numbers ...................... 3
0.14 Sums of products of reciprocals of natural numbers ............... 3
0.15 Sums of the binomial coefficients ......................... 3
0.2 Numerical Series and Infinite Products ...................... 6
0.21 The convergence of numerical series ....................... 6
0.22 Convergence tests ................................. 6
0.23–0.24 Examples of numerical series ........................... 8
0.25 Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.26 Examples of infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.3 Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.30 Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.31 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
0.32 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
0.33 Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
0.4 Certain Formulas from Differential Calculus . . . . . . . . . . . . . . . . . . . . 21
0.41 Differentiation of a definite integral with respect to a parameter . . . . . . . . . 21
0.42 The nth derivative of a product (Leibniz’s rule) . . . . . . . . . . . . . . . . . . 22
0.43 The nth derivative of a composite function . . . . . . . . . . . . . . . . . . . . 22
0.44 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1 Elementary Functions 25
1.1 Power of Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.12 Series of rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
vi CONTENTS
1.21 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.22 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.23 Series of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3–1.4 Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . 28
1.30 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.31 The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.32 The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . . 31
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions . . . . . . . . . . . 33
1.34 Certain sums of trigonometric and hyperbolic functions . . . . . . . . . . . . . 36
1.35 Sums of powers of trigonometric functions of multiple angles . . . . . . . . . . 37
1.36 Sums of products of trigonometric functions of multiple angles . . . . . . . . . 38
1.37 Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . 39
1.38 Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . . 39
1.39 The representation of cosines and sines of multiples of the angle as finite products 41
1.41 The expansion of trigonometric and hyperbolic functions in power series . . . . 42
1.42 Expansion in series of simple fractions . . . . . . . . . . . . . . . . . . . . . . 44
1.43 Representation in the form of an infinite product . . . . . . . . . . . . . . . . . 45
1.44–1.45 Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.46 Series of products of exponential and trigonometric functions . . . . . . . . . . 51
1.47 Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . . 51
1.49 The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . . 52
1.5 The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.51 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.52 Series of logarithms (cf. 1.431) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6 The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . 56
1.61 The domain of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.62–1.63 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.64 Series representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2 Indefinite Integrals of Elementary Functions 63
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.00 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.01 The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.02 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.10 General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.11–2.13 Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . . 68
2.14 Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . . 74
2.15 Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . . 78
2.16 Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . . 78
2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . . 79
2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx 81
2.2 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.20 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.21 Forms containing the binomial a + bxk and √x . . . . . . . . . . . . . . . . . 83
CONTENTS vii
2.22–2.23 Forms containing n (a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.24 Forms containing √a + bx and the binomial α + βx . . . . . . . . . . . . . . . 88
2.25 Forms containing √a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.26 Forms containing √a + bx + cx2 and integral powers of x . . . . . . . . . . . . 94
2.27 Forms containing √a + cx2 and integral powers of x . . . . . . . . . . . . . . . 99
2.28 Forms containing √a + bx + cx2 and first- and second-degree polynomials . . . 103
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals . . . . . . . 104
2.3 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.31 Forms containing eax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.32 The exponential combined with rational functions of x . . . . . . . . . . . . . . 106
2.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . . . 110
2.44–2.45 Rational functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . 125
2.46 Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . 132
2.47 Combinations of hyperbolic functions and powers . . . . . . . . . . . . . . . . 139
2.48 Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . 148
2.5–2.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.50 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.51–2.52 Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated functions
of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.55–2.56 Rational functions of the sine and cosine . . . . . . . . . . . . . . . . . . . . . 171
2.57 Integrals containing √a ± b sin x or √a ± b cos x . . . . . . . . . . . . . . . . . 179
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . . . 184
2.63–2.65 Products of trigonometric functions and powers . . . . . . . . . . . . . . . . . 214
2.66 Combinations of trigonometric functions and exponentials . . . . . . . . . . . . 227
2.67 Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . 231
2.7 Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . . . 237
2.71 The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
2.72–2.73 Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . 238
2.74 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
2.8 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.81 Arcsines and arccosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent . . . . . 242
2.83 Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . . . 242
2.84 Combinations of the arcsecant and arccosecant with powers of x . . . . . . . . 244
2.85 Combinations of the arctangent and arccotangent with algebraic functions . . . 244
3–4 Definite Integrals of Elementary Functions 247
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.01 Theorems of a general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.02 Change of variable in a definite integral . . . . . . . . . . . . . . . . . . . . . . 248
3.03 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
3.04 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
3.05 The principal values of improper integrals . . . . . . . . . . . . . . . . . . . . . 252
3.1–3.2 Power and Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.11 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
viii CONTENTS
3.12 Products of rational functions and expressions that can be reduced to square
roots of first- and second-degree polynomials . . . . . . . . . . . . . . . . . . . 254
3.13–3.17 Expressions that can be reduced to square roots of third- and fourth-degree
polynomials and their products with rational functions . . . . . . . . . . . . . . 254
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions . . . . . . . . . . . . . . . . . . . . . 313
3.19–3.23 Combinations of powers of x and powers of binomials of the form (α + βx) . . 315
3.24–3.27 Powers of x, of binomials of the form α + βxp and of polynomials in x . . . . . 322
3.3–3.4 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
3.31 Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
3.32–3.34 Exponentials of more complicated arguments . . . . . . . . . . . . . . . . . . . 336
3.35 Combinations of exponentials and rational functions . . . . . . . . . . . . . . . 340
3.36–3.37 Combinations of exponentials and algebraic functions . . . . . . . . . . . . . . 344
3.38–3.39 Combinations of exponentials and arbitrary powers . . . . . . . . . . . . . . . . 346
3.41–3.44 Combinations of rational functions of powers and exponentials . . . . . . . . . 353
3.45 Combinations of powers and algebraic functions of exponentials . . . . . . . . . 363
3.46–3.48 Combinations of exponentials of more complicated arguments and powers . . . 364
3.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
3.51 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
3.52–3.53 Combinations of hyperbolic functions and algebraic functions . . . . . . . . . . 375
3.54 Combinations of hyperbolic functions and exponentials . . . . . . . . . . . . . 382
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . 386
3.6–4.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
3.62 Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 395
3.63 Powers of trigonometric functions and trigonometric functions of linear functions 397
3.64–3.65 Powers and rational functions of trigonometric functions . . . . . . . . . . . . . 401
3.66 Forms containing powers of linear functions of trigonometric functions . . . . . 405
3.67 Square roots of expressions containing trigonometric functions . . . . . . . . . 408
3.68 Various forms of powers of trigonometric functions . . . . . . . . . . . . . . . . 411
3.69–3.71 Trigonometric functions of more complicated arguments . . . . . . . . . . . . . 415
3.72–3.74 Combinations of trigonometric and rational functions . . . . . . . . . . . . . . 423
3.75 Combinations of trigonometric and algebraic functions . . . . . . . . . . . . . . 434
3.76–3.77 Combinations of trigonometric functions and powers . . . . . . . . . . . . . . . 436
3.78–3.81 Rational functions of x and of trigonometric functions . . . . . . . . . . . . . . 447
3.82–3.83 Powers of trigonometric functions combined with other powers . . . . . . . . . 459
3.84 Integrals containing 1 − k2 sin2 x,
√1 − k2 cos2 x, and similar expressions . . 472
3.85–3.88 Trigonometric functions of more complicated arguments combined with powers 475
3.89–3.91 Trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . 485
3.92 Trigonometric functions of more complicated arguments combined with exponentials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
3.93 Trigonometric and exponential functions of trigonometric functions . . . . . . . 495
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers . . . 497
3.98–3.99 Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . 509
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers . . . 516
4.13 Combinations of trigonometric and hyperbolic functions and exponentials . . . . 522
CONTENTS ix
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 525
4.2–4.4 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
4.21 Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
4.22 Logarithms of more complicated arguments . . . . . . . . . . . . . . . . . . . . 529
4.23 Combinations of logarithms and rational functions . . . . . . . . . . . . . . . . 535
4.24 Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . 538
4.25 Combinations of logarithms and powers . . . . . . . . . . . . . . . . . . . . . . 540
4.26–4.27 Combinations involving powers of the logarithm and other powers . . . . . . . . 542
4.28 Combinations of rational functions of ln x and powers . . . . . . . . . . . . . . 553
4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 555
4.33–4.34 Combinations of logarithms and exponentials . . . . . . . . . . . . . . . . . . . 571
4.35–4.36 Combinations of logarithms, exponentials, and powers . . . . . . . . . . . . . . 573
4.37 Combinations of logarithms and hyperbolic functions . . . . . . . . . . . . . . . 578
4.38–4.41 Logarithms and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 581
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers . . . . . . . . 594
4.44 Combinations of logarithms, trigonometric functions, and exponentials . . . . . 599
4.5 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 599
4.51 Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 599
4.52 Combinations of arcsines, arccosines, and powers . . . . . . . . . . . . . . . . . 600
4.53–4.54 Combinations of arctangents, arccotangents, and powers . . . . . . . . . . . . . 601
4.55 Combinations of inverse trigonometric functions and exponentials . . . . . . . . 605
4.56 A combination of the arctangent and a hyperbolic function . . . . . . . . . . . 605
4.57 Combinations of inverse and direct trigonometric functions . . . . . . . . . . . 605
4.58 A combination involving an inverse and a direct trigonometric function and a
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
4.59 Combinations of inverse trigonometric functions and logarithms . . . . . . . . . 607
4.6 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
4.60 Change of variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . 607
4.61 Change of the order of integration and change of variables . . . . . . . . . . . 608
4.62 Double and triple integrals with constant limits . . . . . . . . . . . . . . . . . . 610
4.63–4.64 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
5 Indefinite Integrals of Special Functions 619
5.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 619
5.11 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
5.12 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
5.13 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
5.14 Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
5.2 The Exponential Integral Function . . . . . . . . . . . . . . . . . . . . . . . . 627
5.21 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . . 627
5.22 Combinations of the exponential integral function and powers . . . . . . . . . . 627
5.23 Combinations of the exponential integral and the exponential . . . . . . . . . . 628
5.3 The Sine Integral and the Cosine Integral . . . . . . . . . . . . . . . . . . . . . 628
5.4 The Probability Integral and Fresnel Integrals . . . . . . . . . . . . . . . . . . . 629
5.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
x CONTENTS
6–7 Definite Integrals of Special Functions 631
6.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6.11 Forms containing F(x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6.12 Forms containing E(x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
6.13 Integration of elliptic integrals with respect to the modulus . . . . . . . . . . . 632
6.14–6.15 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
6.16 The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
6.17 Generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
6.2–6.3 The Exponential Integral Function and Functions Generated by It . . . . . . . . 636
6.21 The logarithm integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
6.22–6.23 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . . 638
6.24–6.26 The sine integral and cosine integral functions . . . . . . . . . . . . . . . . . . 639
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions . . . . . . 644
6.28–6.31 The probability integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
6.32 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
6.4 The Gamma Function and Functions Generated by It . . . . . . . . . . . . . . 650
6.41 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
6.42 Combinations of the gamma function, the exponential, and powers . . . . . . . 652
6.43 Combinations of the gamma function and trigonometric functions . . . . . . . . 655
6.44 The logarithm of the gamma function∗ . . . . . . . . . . . . . . . . . . . . . . 656
6.45 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 657
6.46–6.47 The function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
6.5–6.7 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
6.51 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
6.52 Bessel functions combined with x and x2 . . . . . . . . . . . . . . . . . . . . . 664
6.53–6.54 Combinations of Bessel functions and rational functions . . . . . . . . . . . . . 670
6.55 Combinations of Bessel functions and algebraic functions . . . . . . . . . . . . 674
6.56–6.58 Combinations of Bessel functions and powers . . . . . . . . . . . . . . . . . . . 675
6.59 Combinations of powers and Bessel functions of more complicated arguments . 689
6.61 Combinations of Bessel functions and exponentials . . . . . . . . . . . . . . . . 694
6.62–6.63 Combinations of Bessel functions, exponentials, and powers . . . . . . . . . . . 699
6.64 Combinations of Bessel functions of more complicated arguments, exponentials,
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
6.65 Combinations of Bessel and exponential functions of more complicated arguments
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
6.66 Combinations of Bessel, hyperbolic, and exponential functions . . . . . . . . . . 713
6.67–6.68 Combinations of Bessel and trigonometric functions . . . . . . . . . . . . . . . 717
6.69–6.74 Combinations of Bessel and trigonometric functions and powers . . . . . . . . . 727
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers . 742
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions . . . . . . . . 747
6.77 Combinations of Bessel functions and the logarithm, or arctangent . . . . . . . 747
6.78 Combinations of Bessel and other special functions . . . . . . . . . . . . . . . . 748
6.79 Integration of Bessel functions with respect to the order . . . . . . . . . . . . . 749
6.8 Functions Generated by Bessel Functions . . . . . . . . . . . . . . . . . . . . . 753
6.81 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
6.82 Combinations of Struve functions, exponentials, and powers . . . . . . . . . . . 754
6.83 Combinations of Struve and trigonometric functions . . . . . . . . . . . . . . . 755
CONTENTS xi
6.84–6.85 Combinations of Struve and Bessel functions . . . . . . . . . . . . . . . . . . . 756
6.86 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
6.87 Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
6.9 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
6.91 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions . . . . . . . 763
6.93 Combinations of Mathieu and Bessel functions . . . . . . . . . . . . . . . . . . 767
6.94 Relationships between eigenfunctions of the Helmholtz equation in different
coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
7.1–7.2 Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 769
7.11 Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
7.12–7.13 Combinations of associated Legendre functions and powers . . . . . . . . . . . 770
7.14 Combinations of associated Legendre functions, exponentials, and powers . . . 776
7.15 Combinations of associated Legendre and hyperbolic functions . . . . . . . . . 778
7.16 Combinations of associated Legendre functions, powers, and trigonometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
7.17 A combination of an associated Legendre function and the probability integral . 781
7.18 Combinations of associated Legendre and Bessel functions . . . . . . . . . . . . 782
7.19 Combinations of associated Legendre functions and functions generated by
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
7.21 Integration of associated Legendre functions with respect to the order . . . . . 788
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 789
7.23 Combinations of Legendre polynomials and powers . . . . . . . . . . . . . . . . 791
7.24 Combinations of Legendre polynomials and other elementary functions . . . . . 792
7.25 Combinations of Legendre polynomials and Bessel functions . . . . . . . . . . . 794
7.3–7.4 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
7.31 Combinations of Gegenbauer polynomials Cν
n(x) and powers . . . . . . . . . . 795
7.32 Combinations of Gegenbauer polynomials Cν
n(x) and elementary functions . . . 797
7.325∗ Complete System of Orthogonal Step Functions . . . . . . . . . . . . . . . . . 798
7.33 Combinations of the polynomials C ν
n(x) and Bessel functions; Integration of
Gegenbauer functions with respect to the index . . . . . . . . . . . . . . . . . 798
7.34 Combinations of Chebyshev polynomials and powers . . . . . . . . . . . . . . . 800
7.35 Combinations of Chebyshev polynomials and elementary functions . . . . . . . 802
7.36 Combinations of Chebyshev polynomials and Bessel functions . . . . . . . . . . 803
7.37–7.38 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
7.39 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
7.41–7.42 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808
7.5 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812
7.51 Combinations of hypergeometric functions and powers . . . . . . . . . . . . . . 812
7.52 Combinations of hypergeometric functions and exponentials . . . . . . . . . . . 814
7.53 Hypergeometric and trigonometric functions . . . . . . . . . . . . . . . . . . . 817
7.54 Combinations of hypergeometric and Bessel functions . . . . . . . . . . . . . . 817
7.6 Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 820
7.61 Combinations of confluent hypergeometric functions and powers . . . . . . . . 820
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials . . . . . 822
7.64 Combinations of confluent hypergeometric and trigonometric functions . . . . . 829
7.65 Combinations of confluent hypergeometric functions and Bessel functions . . . 830
xii CONTENTS
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 831
7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials,
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
7.68 Combinations of confluent hypergeometric functions and other special functions 839
7.69 Integration of confluent hypergeometric functions with respect to the index . . 841
7.7 Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
7.71 Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
7.72 Combinations of parabolic cylinder functions, powers, and exponentials . . . . . 842
7.73 Combinations of parabolic cylinder and hyperbolic functions . . . . . . . . . . . 843
7.74 Combinations of parabolic cylinder and trigonometric functions . . . . . . . . . 844
7.75 Combinations of parabolic cylinder and Bessel functions . . . . . . . . . . . . . 845
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849
7.77 Integration of a parabolic cylinder function with respect to the index . . . . . . 849
7.8 Meijer’s and MacRobert’s Functions (G and E) . . . . . . . . . . . . . . . . . 850
7.81 Combinations of the functions G and E and the elementary functions . . . . . 850
7.82 Combinations of the functions G and E and Bessel functions . . . . . . . . . . 854
7.83 Combinations of the functions G and E and other special functions . . . . . . . 856
8–9 Special Functions 859
8.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 859
8.11 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
8.12 Functional relations between elliptic integrals . . . . . . . . . . . . . . . . . . . 863
8.13 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865
8.14 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866
8.15 Properties of Jacobian elliptic functions and functional relationships between them 870
8.16 The Weierstrass function ℘(u) . . . . . . . . . . . . . . . . . . . . . . . . . . 873
8.17 The functions ζ(u) and σ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . 876
8.18–8.19 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
8.2 The Exponential Integral Function and Functions Generated by It . . . . . . . . 883
8.21 The exponential integral function Ei(x) . . . . . . . . . . . . . . . . . . . . . . 883
8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x . . . 886
8.23 The sine integral and the cosine integral: si x and ci x . . . . . . . . . . . . . . 886
8.24 The logarithm integral li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
8.25 The probability integral Φ(x), the Fresnel integrals S(x) and C (x), the error
function erf(x), and the complementary error function erfc(x) . . . . . . . . . 887
8.26 Lobachevskiy’s function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 891
8.3 Euler’s Integrals of the First and Second Kinds . . . . . . . . . . . . . . . . . . 892
8.31 The gamma function (Euler’s integral of the second kind): Γ(z) . . . . . . . . 892
8.32 Representation of the gamma function as series and products . . . . . . . . . . 894
8.33 Functional relations involving the gamma function . . . . . . . . . . . . . . . . 895
8.34 The logarithm of the gamma function . . . . . . . . . . . . . . . . . . . . . . . 898
8.35 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 899
8.36 The psi function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902
8.37 The function β(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906
8.38 The beta function (Euler’s integral of the first kind): B(x, y) . . . . . . . . . . 908
8.39 The incomplete beta function Bx(p, q) . . . . . . . . . . . . . . . . . . . . . . 910
8.4–8.5 Bessel Functions and Functions Associated with Them . . . . . . . . . . . . . . 910
CONTENTS xiii
8.40 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910
8.41 Integral representations of the functions Jν(z) and Nν(z) . . . . . . . . . . . . 912
8.42 Integral representations of the functions H(1)
ν (z) and H(2)
ν (z) . . . . . . . . . . 914
8.43 Integral representations of the functions Iν(z) and Kν(z) . . . . . . . . . . . . 916
8.44 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918
8.45 Asymptotic expansions of Bessel functions . . . . . . . . . . . . . . . . . . . . 920
8.46 Bessel functions of order equal to an integer plus one-half . . . . . . . . . . . . 924
8.47–8.48 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
8.49 Differential equations leading to Bessel functions . . . . . . . . . . . . . . . . . 931
8.51–8.52 Series of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933
8.53 Expansion in products of Bessel functions . . . . . . . . . . . . . . . . . . . . . 940
8.54 The zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
8.55 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
8.56 Thomson functions and their generalizations . . . . . . . . . . . . . . . . . . . 944
8.57 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945
8.58 Anger and Weber functions Jν(z) and Eν(z) . . . . . . . . . . . . . . . . . . . 948
8.59 Neumann’s and Schl¨afli’s polynomials: On(z) and S n(z) . . . . . . . . . . . . 949
8.6 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
8.60 Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
8.61 Periodic Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
8.62 Recursion relations for the coefficients A(2n)
2r , A(2n+1)
2r+1 , B(2n+1)
2r+1 , B(2n+2)
2r+2 . . . . 951
8.63 Mathieu functions with a purely imaginary argument . . . . . . . . . . . . . . . 952
8.64 Non-periodic solutions of Mathieu’s equation . . . . . . . . . . . . . . . . . . . 953
8.65 Mathieu functions for negative q . . . . . . . . . . . . . . . . . . . . . . . . . 953
8.66 Representation of Mathieu functions as series of Bessel functions . . . . . . . . 954
8.67 The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
8.7–8.8 Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 958
8.70 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958
8.71 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
8.72 Asymptotic series for large values of |ν| . . . . . . . . . . . . . . . . . . . . . . 962
8.73–8.74 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964
8.75 Special cases and particular values . . . . . . . . . . . . . . . . . . . . . . . . 968
8.76 Derivatives with respect to the order . . . . . . . . . . . . . . . . . . . . . . . 969
8.77 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
8.78 The zeros of associated Legendre functions . . . . . . . . . . . . . . . . . . . . 972
8.79 Series of associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . 972
8.81 Associated Legendre functions with integer indices . . . . . . . . . . . . . . . . 974
8.82–8.83 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975
8.84 Conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980
8.85 Toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981
8.9 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
8.90 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
8.91 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983
8.919 Series of products of Legendre and Chebyshev polynomials . . . . . . . . . . . 988
8.92 Series of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 988
8.93 Gegenbauer polynomials Cλ
n (t) . . . . . . . . . . . . . . . . . . . . . . . . . . 990
8.94 The Chebyshev polynomials Tn(x) and Un(x) . . . . . . . . . . . . . . . . . . 993
xiv CONTENTS
8.95 The Hermite polynomials H n(x) . . . . . . . . . . . . . . . . . . . . . . . . . 996
8.96 Jacobi’s polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998
8.97 The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000
9.1 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.10 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.11 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.12 Representation of elementary functions in terms of a hypergeometric functions . 1006
9.13 Transformation formulas and the analytic continuation of functions defined by
hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008
9.14 A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . 1010
9.15 The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . 1010
9.16 Riemann’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
9.17 Representing the solutions to certain second-order differential equations using
a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017
9.18 Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . . 1018
9.19 A hypergeometric function of several variables . . . . . . . . . . . . . . . . . . 1022
9.2 Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 1022
9.20 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
9.21 The functions Φ(α,γ; z) and Ψ(α,γ; z) . . . . . . . . . . . . . . . . . . . . . . 1023
9.22–9.23 The Whittaker functions Mλ,μ(z) and Wλ,μ(z) . . . . . . . . . . . . . . . . . . 1024
9.24–9.25 Parabolic cylinder functions Dp(z) . . . . . . . . . . . . . . . . . . . . . . . . 1028
9.26 Confluent hypergeometric series of two variables . . . . . . . . . . . . . . . . . 1031
9.3 Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.30 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.31 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033
9.32 A differential equation for the G-function . . . . . . . . . . . . . . . . . . . . . 1034
9.33 Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034
9.34 Connections with other special functions . . . . . . . . . . . . . . . . . . . . . 1034
9.4 MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.41 Representation by means of multiple integrals . . . . . . . . . . . . . . . . . . 1035
9.42 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.5 Riemann’s Zeta Functions ζ(z,q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) 1036
9.51 Definition and integral representations . . . . . . . . . . . . . . . . . . . . . . 1036
9.52 Representation as a series or as an infinite product . . . . . . . . . . . . . . . . 1037
9.53 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
9.54 Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038
9.55 The Lerch function Φ(z, s, v) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039
9.56 The function ξ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.6 Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . . 1040
9.61 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.62 Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041
9.63 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
9.64 The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) . . . . . . . . . . 1043
9.65 Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044
9.7 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.71 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.72 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
CONTENTS xv
9.73 Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
9.74 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
10 Vector Field Theory 1049
10.1–10.8 Vectors, Vector Operators, and Integral Theorems . . . . . . . . . . . . . . . . 1049
10.11 Products of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.12 Properties of scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.13 Properties of vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.14 Differentiation of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.21 Operators grad, div, and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.31 Properties of the operator ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051
10.41 Solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.51–10.61 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.71–10.72 Vector integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
10.81 Integral rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . . . 1057
11 Algebraic Inequalities 1059
11.1–11.3 General Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059
11.11 Algebraic inequalities involving real numbers . . . . . . . . . . . . . . . . . . . 1059
11.21 Algebraic inequalities involving complex numbers . . . . . . . . . . . . . . . . . 1060
11.31 Inequalities for sets of complex numbers . . . . . . . . . . . . . . . . . . . . . 1061
12 Integral Inequalities 1063
12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.111 First mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.112 Second mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.113 First mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . . 1063
12.114 Second mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . 1064
12.21 Differentiation of Definite Integral Containing a Parameter . . . . . . . . . . . 1064
12.211 Differentiation when limits are finite . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.212 Differentiation when a limit is infinite . . . . . . . . . . . . . . . . . . . . . . . 1064
12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . . . . . . . . . . . . 1064
12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.316 Steffensen’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.51 Fourier Series and Related Inequalities . . . . . . . . . . . . . . . . . . . . . . 1066
12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.513 Parseval’s theorem for trigonometric Fourier series . . . . . . . . . . . . . . . . 1067
12.514 Integral representation of the nth partial sum . . . . . . . . . . . . . . . . . . . 1067
xvi CONTENTS
12.515 Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.516 Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
12.517 Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
13 Matrices and Related Results 1069
13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.129 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.130 Non-negative definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.214 Positive definite and semidefinite quadratic form . . . . . . . . . . . . . . . . . 1072
13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.31 Differentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073
13.41 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
14 Determinants 1075
14.11 Expansion of Second- and Third-Order Determinants . . . . . . . . . . . . . . 1075
14.12 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075
14.13 Minors and Cofactors of a Determinant . . . . . . . . . . . . . . . . . . . . . . 1075
14.14 Principal Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.15* Laplace Expansion of a Determinant . . . . . . . . . . . . . . . . . . . . . . . 1076
14.16 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.17 Hadamard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.18 Hadamard’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.21 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.31 Some Special Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
CONTENTS xvii
14.311 Vandermonde’s determinant (alternant) . . . . . . . . . . . . . . . . . . . . . . 1078
14.312 Circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.313 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.314 Hessian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.315 Wronskian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.316 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.317 Gram-Kowalewski theorem on linear dependence . . . . . . . . . . . . . . . . . 1080
15 Norms 1081
15.1–15.9 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.11 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.21 Principal Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.211 The norm ||x||1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.212 The norm ||x||2 (Euclidean or L2 norm) . . . . . . . . . . . . . . . . . . . . . 1081
15.213 The norm ||x||∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.31 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.311 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.312 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.313 Natural norm of unit matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.41 Principal Natural Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.411 Maximum absolute column sum norm . . . . . . . . . . . . . . . . . . . . . . . 1082
15.412 Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.413 Maximum absolute row sum norm . . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.51 Spectral Radius of a Square Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.511 Inequalities concerning matrix norms and the spectral radius . . . . . . . . . . 1083
15.512 Deductions from Gerschgorin’s theorem (see 15.814) . . . . . . . . . . . . . . 1083
15.61 Inequalities Involving Eigenvalues of Matrices . . . . . . . . . . . . . . . . . . . 1084
15.611 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.612 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.71 Inequalities for the Characteristic Polynomial . . . . . . . . . . . . . . . . . . . 1084
15.711 Named and unnamed inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1085
15.712 Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.713 Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.714 Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.715 Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.81–15.82 Named Theorems on Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.811 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.812 Sturmian separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.813 Poincare’s separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.814 Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.815 Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.816 Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.817 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.818 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.819 Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.820 Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.821 First theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.822 Second theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . 1089
xviii CONTENTS
15.823 Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov . . . . . . . . . . . . . . . . . . 1089
15.91 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.911 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.912 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
16 Ordinary Differential Equations 1093
16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093
16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093
16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094
16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094
16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096
16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096
16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096
16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097
16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.516 Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100
16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100
16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.626 Szeg¨o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
CONTENTS xix
16.712 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.81–16.82 Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.811 Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.822 Comparison theorem for non-oscillation . . . . . . . . . . . . . . . . . . . . . . 1104
16.823 Necessary and sufficient conditions for non-oscillation . . . . . . . . . . . . . . 1104
16.91 Some Growth Estimates for Solutions of Second-Order Equations . . . . . . . . 1104
16.911 Strictly increasing and decreasing solutions . . . . . . . . . . . . . . . . . . . . 1104
16.912 General result on dominant and subdominant solutions . . . . . . . . . . . . . 1104
16.913 Estimate of dominant solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.914 A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.92 Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.921 All solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.922 If all solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.923 If a(x) → ∞ monotonically as x → ∞, then all solutions of . . . . . . . . . . . 1106
16.924 Consider the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.93 Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
17 Fourier, Laplace, and Mellin Transforms 1107
17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . . . . . . . . 1107
17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . 1108
17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . . 1118
17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
17.24 Table of Fourier transform pairs for spherically symmetric functions . . . . . . . 1120
17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1121
17.32 Basic properties of the Fourier sine and cosine transforms . . . . . . . . . . . . 1121
17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 1122
17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1126
17.35 Relationships between transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . . . . . . . . 1130
17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131
18 The z-Transform 1135
18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms . . . . . . . . . . . . . . . . . 1135
18.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
18.2 Bilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136
18.3 Unilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
References 1141
Supplemental references 1145
Index of Functions and Constants 1151
General Index of Concep
|
| Côte titre : |
Fs/12411,Fs/10427 |
Table of integrals, series and products [texte imprimé] / Gradshtein, Izrail Solomonovich, Auteur ; Iosif Moiseevich Ryzhik, Auteur . - 7e ed. . - Amsterdam : Elsevier, 2007 . - 1 vol (1171 p.) : ill. ; 25 cm + 1 CD-ROM (sd., col. ; 4 3/4 in.). ISBN : 978-0-12-373637-6 978-0-12-373637-6 Langues : Anglais ( eng) Langues originales : Anglais ( eng)
| Catégories : |
Mathématique
|
| Mots-clés : |
Intégrales : Tables
Mathématiques : Tables
Séries (mathématiques) : Tables |
| Index. décimale : |
515-Analyse mathèmatique |
| Résumé : |
Le tableau des intégrales, des séries et des produits est la référence essentielle pour les intégrales en anglais. Les mathématiciens, les scientifiques et les ingénieurs s'en remettent à l'identification et à la résolution de problèmes extrêmement complexes. Depuis la publication de la première édition en langue anglaise en 1965, elle a été entièrement révisée et agrandie régulièrement, avec des ajouts importants et, le cas échéant, des écritures existantes corrigées ou révisées. La septième édition comprend un CD-Rom entièrement consultable. |
| Note de contenu : |
Sommaire
0 Introduction 1
0.1 Finite Sums ..................................... 1
0.11 Progressions .................................... 1
0.12 Sums of powers of natural numbers ........................ 1
0.13 Sums of reciprocals of natural numbers ...................... 3
0.14 Sums of products of reciprocals of natural numbers ............... 3
0.15 Sums of the binomial coefficients ......................... 3
0.2 Numerical Series and Infinite Products ...................... 6
0.21 The convergence of numerical series ....................... 6
0.22 Convergence tests ................................. 6
0.23–0.24 Examples of numerical series ........................... 8
0.25 Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.26 Examples of infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.3 Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.30 Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.31 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
0.32 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
0.33 Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
0.4 Certain Formulas from Differential Calculus . . . . . . . . . . . . . . . . . . . . 21
0.41 Differentiation of a definite integral with respect to a parameter . . . . . . . . . 21
0.42 The nth derivative of a product (Leibniz’s rule) . . . . . . . . . . . . . . . . . . 22
0.43 The nth derivative of a composite function . . . . . . . . . . . . . . . . . . . . 22
0.44 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1 Elementary Functions 25
1.1 Power of Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.12 Series of rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
vi CONTENTS
1.21 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.22 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.23 Series of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3–1.4 Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . 28
1.30 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.31 The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.32 The representation of powers of trigonometric and hyperbolic functions in terms
of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . . 31
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions . . . . . . . . . . . 33
1.34 Certain sums of trigonometric and hyperbolic functions . . . . . . . . . . . . . 36
1.35 Sums of powers of trigonometric functions of multiple angles . . . . . . . . . . 37
1.36 Sums of products of trigonometric functions of multiple angles . . . . . . . . . 38
1.37 Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . 39
1.38 Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . . 39
1.39 The representation of cosines and sines of multiples of the angle as finite products 41
1.41 The expansion of trigonometric and hyperbolic functions in power series . . . . 42
1.42 Expansion in series of simple fractions . . . . . . . . . . . . . . . . . . . . . . 44
1.43 Representation in the form of an infinite product . . . . . . . . . . . . . . . . . 45
1.44–1.45 Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.46 Series of products of exponential and trigonometric functions . . . . . . . . . . 51
1.47 Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . . 51
1.49 The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . . 52
1.5 The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.51 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.52 Series of logarithms (cf. 1.431) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6 The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . 56
1.61 The domain of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.62–1.63 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.64 Series representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2 Indefinite Integrals of Elementary Functions 63
2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.00 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.01 The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.02 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.10 General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.11–2.13 Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . . 68
2.14 Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . . 74
2.15 Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . . 78
2.16 Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . . 78
2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . . 79
2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx 81
2.2 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.20 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.21 Forms containing the binomial a + bxk and √x . . . . . . . . . . . . . . . . . 83
CONTENTS vii
2.22–2.23 Forms containing n (a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.24 Forms containing √a + bx and the binomial α + βx . . . . . . . . . . . . . . . 88
2.25 Forms containing √a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.26 Forms containing √a + bx + cx2 and integral powers of x . . . . . . . . . . . . 94
2.27 Forms containing √a + cx2 and integral powers of x . . . . . . . . . . . . . . . 99
2.28 Forms containing √a + bx + cx2 and first- and second-degree polynomials . . . 103
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals . . . . . . . 104
2.3 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.31 Forms containing eax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.32 The exponential combined with rational functions of x . . . . . . . . . . . . . . 106
2.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . . . 110
2.44–2.45 Rational functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . 125
2.46 Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . 132
2.47 Combinations of hyperbolic functions and powers . . . . . . . . . . . . . . . . 139
2.48 Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . 148
2.5–2.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.50 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.51–2.52 Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated functions
of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.55–2.56 Rational functions of the sine and cosine . . . . . . . . . . . . . . . . . . . . . 171
2.57 Integrals containing √a ± b sin x or √a ± b cos x . . . . . . . . . . . . . . . . . 179
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . . . 184
2.63–2.65 Products of trigonometric functions and powers . . . . . . . . . . . . . . . . . 214
2.66 Combinations of trigonometric functions and exponentials . . . . . . . . . . . . 227
2.67 Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . 231
2.7 Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . . . 237
2.71 The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
2.72–2.73 Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . 238
2.74 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
2.8 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.81 Arcsines and arccosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent . . . . . 242
2.83 Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . . . 242
2.84 Combinations of the arcsecant and arccosecant with powers of x . . . . . . . . 244
2.85 Combinations of the arctangent and arccotangent with algebraic functions . . . 244
3–4 Definite Integrals of Elementary Functions 247
3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.01 Theorems of a general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.02 Change of variable in a definite integral . . . . . . . . . . . . . . . . . . . . . . 248
3.03 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
3.04 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
3.05 The principal values of improper integrals . . . . . . . . . . . . . . . . . . . . . 252
3.1–3.2 Power and Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.11 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
viii CONTENTS
3.12 Products of rational functions and expressions that can be reduced to square
roots of first- and second-degree polynomials . . . . . . . . . . . . . . . . . . . 254
3.13–3.17 Expressions that can be reduced to square roots of third- and fourth-degree
polynomials and their products with rational functions . . . . . . . . . . . . . . 254
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions . . . . . . . . . . . . . . . . . . . . . 313
3.19–3.23 Combinations of powers of x and powers of binomials of the form (α + βx) . . 315
3.24–3.27 Powers of x, of binomials of the form α + βxp and of polynomials in x . . . . . 322
3.3–3.4 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
3.31 Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
3.32–3.34 Exponentials of more complicated arguments . . . . . . . . . . . . . . . . . . . 336
3.35 Combinations of exponentials and rational functions . . . . . . . . . . . . . . . 340
3.36–3.37 Combinations of exponentials and algebraic functions . . . . . . . . . . . . . . 344
3.38–3.39 Combinations of exponentials and arbitrary powers . . . . . . . . . . . . . . . . 346
3.41–3.44 Combinations of rational functions of powers and exponentials . . . . . . . . . 353
3.45 Combinations of powers and algebraic functions of exponentials . . . . . . . . . 363
3.46–3.48 Combinations of exponentials of more complicated arguments and powers . . . 364
3.5 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
3.51 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
3.52–3.53 Combinations of hyperbolic functions and algebraic functions . . . . . . . . . . 375
3.54 Combinations of hyperbolic functions and exponentials . . . . . . . . . . . . . 382
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . 386
3.6–4.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
3.62 Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 395
3.63 Powers of trigonometric functions and trigonometric functions of linear functions 397
3.64–3.65 Powers and rational functions of trigonometric functions . . . . . . . . . . . . . 401
3.66 Forms containing powers of linear functions of trigonometric functions . . . . . 405
3.67 Square roots of expressions containing trigonometric functions . . . . . . . . . 408
3.68 Various forms of powers of trigonometric functions . . . . . . . . . . . . . . . . 411
3.69–3.71 Trigonometric functions of more complicated arguments . . . . . . . . . . . . . 415
3.72–3.74 Combinations of trigonometric and rational functions . . . . . . . . . . . . . . 423
3.75 Combinations of trigonometric and algebraic functions . . . . . . . . . . . . . . 434
3.76–3.77 Combinations of trigonometric functions and powers . . . . . . . . . . . . . . . 436
3.78–3.81 Rational functions of x and of trigonometric functions . . . . . . . . . . . . . . 447
3.82–3.83 Powers of trigonometric functions combined with other powers . . . . . . . . . 459
3.84 Integrals containing 1 − k2 sin2 x,
√1 − k2 cos2 x, and similar expressions . . 472
3.85–3.88 Trigonometric functions of more complicated arguments combined with powers 475
3.89–3.91 Trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . 485
3.92 Trigonometric functions of more complicated arguments combined with exponentials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
3.93 Trigonometric and exponential functions of trigonometric functions . . . . . . . 495
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers . . . 497
3.98–3.99 Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . 509
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers . . . 516
4.13 Combinations of trigonometric and hyperbolic functions and exponentials . . . . 522
CONTENTS ix
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 525
4.2–4.4 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
4.21 Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
4.22 Logarithms of more complicated arguments . . . . . . . . . . . . . . . . . . . . 529
4.23 Combinations of logarithms and rational functions . . . . . . . . . . . . . . . . 535
4.24 Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . 538
4.25 Combinations of logarithms and powers . . . . . . . . . . . . . . . . . . . . . . 540
4.26–4.27 Combinations involving powers of the logarithm and other powers . . . . . . . . 542
4.28 Combinations of rational functions of ln x and powers . . . . . . . . . . . . . . 553
4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 555
4.33–4.34 Combinations of logarithms and exponentials . . . . . . . . . . . . . . . . . . . 571
4.35–4.36 Combinations of logarithms, exponentials, and powers . . . . . . . . . . . . . . 573
4.37 Combinations of logarithms and hyperbolic functions . . . . . . . . . . . . . . . 578
4.38–4.41 Logarithms and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 581
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers . . . . . . . . 594
4.44 Combinations of logarithms, trigonometric functions, and exponentials . . . . . 599
4.5 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 599
4.51 Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 599
4.52 Combinations of arcsines, arccosines, and powers . . . . . . . . . . . . . . . . . 600
4.53–4.54 Combinations of arctangents, arccotangents, and powers . . . . . . . . . . . . . 601
4.55 Combinations of inverse trigonometric functions and exponentials . . . . . . . . 605
4.56 A combination of the arctangent and a hyperbolic function . . . . . . . . . . . 605
4.57 Combinations of inverse and direct trigonometric functions . . . . . . . . . . . 605
4.58 A combination involving an inverse and a direct trigonometric function and a
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
4.59 Combinations of inverse trigonometric functions and logarithms . . . . . . . . . 607
4.6 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
4.60 Change of variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . 607
4.61 Change of the order of integration and change of variables . . . . . . . . . . . 608
4.62 Double and triple integrals with constant limits . . . . . . . . . . . . . . . . . . 610
4.63–4.64 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
5 Indefinite Integrals of Special Functions 619
5.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 619
5.11 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
5.12 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
5.13 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
5.14 Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
5.2 The Exponential Integral Function . . . . . . . . . . . . . . . . . . . . . . . . 627
5.21 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . . 627
5.22 Combinations of the exponential integral function and powers . . . . . . . . . . 627
5.23 Combinations of the exponential integral and the exponential . . . . . . . . . . 628
5.3 The Sine Integral and the Cosine Integral . . . . . . . . . . . . . . . . . . . . . 628
5.4 The Probability Integral and Fresnel Integrals . . . . . . . . . . . . . . . . . . . 629
5.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
x CONTENTS
6–7 Definite Integrals of Special Functions 631
6.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6.11 Forms containing F(x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6.12 Forms containing E(x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
6.13 Integration of elliptic integrals with respect to the modulus . . . . . . . . . . . 632
6.14–6.15 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
6.16 The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
6.17 Generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
6.2–6.3 The Exponential Integral Function and Functions Generated by It . . . . . . . . 636
6.21 The logarithm integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
6.22–6.23 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . . 638
6.24–6.26 The sine integral and cosine integral functions . . . . . . . . . . . . . . . . . . 639
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions . . . . . . 644
6.28–6.31 The probability integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
6.32 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
6.4 The Gamma Function and Functions Generated by It . . . . . . . . . . . . . . 650
6.41 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
6.42 Combinations of the gamma function, the exponential, and powers . . . . . . . 652
6.43 Combinations of the gamma function and trigonometric functions . . . . . . . . 655
6.44 The logarithm of the gamma function∗ . . . . . . . . . . . . . . . . . . . . . . 656
6.45 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 657
6.46–6.47 The function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
6.5–6.7 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
6.51 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
6.52 Bessel functions combined with x and x2 . . . . . . . . . . . . . . . . . . . . . 664
6.53–6.54 Combinations of Bessel functions and rational functions . . . . . . . . . . . . . 670
6.55 Combinations of Bessel functions and algebraic functions . . . . . . . . . . . . 674
6.56–6.58 Combinations of Bessel functions and powers . . . . . . . . . . . . . . . . . . . 675
6.59 Combinations of powers and Bessel functions of more complicated arguments . 689
6.61 Combinations of Bessel functions and exponentials . . . . . . . . . . . . . . . . 694
6.62–6.63 Combinations of Bessel functions, exponentials, and powers . . . . . . . . . . . 699
6.64 Combinations of Bessel functions of more complicated arguments, exponentials,
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
6.65 Combinations of Bessel and exponential functions of more complicated arguments
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
6.66 Combinations of Bessel, hyperbolic, and exponential functions . . . . . . . . . . 713
6.67–6.68 Combinations of Bessel and trigonometric functions . . . . . . . . . . . . . . . 717
6.69–6.74 Combinations of Bessel and trigonometric functions and powers . . . . . . . . . 727
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers . 742
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions . . . . . . . . 747
6.77 Combinations of Bessel functions and the logarithm, or arctangent . . . . . . . 747
6.78 Combinations of Bessel and other special functions . . . . . . . . . . . . . . . . 748
6.79 Integration of Bessel functions with respect to the order . . . . . . . . . . . . . 749
6.8 Functions Generated by Bessel Functions . . . . . . . . . . . . . . . . . . . . . 753
6.81 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
6.82 Combinations of Struve functions, exponentials, and powers . . . . . . . . . . . 754
6.83 Combinations of Struve and trigonometric functions . . . . . . . . . . . . . . . 755
CONTENTS xi
6.84–6.85 Combinations of Struve and Bessel functions . . . . . . . . . . . . . . . . . . . 756
6.86 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
6.87 Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
6.9 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
6.91 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions . . . . . . . 763
6.93 Combinations of Mathieu and Bessel functions . . . . . . . . . . . . . . . . . . 767
6.94 Relationships between eigenfunctions of the Helmholtz equation in different
coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
7.1–7.2 Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 769
7.11 Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
7.12–7.13 Combinations of associated Legendre functions and powers . . . . . . . . . . . 770
7.14 Combinations of associated Legendre functions, exponentials, and powers . . . 776
7.15 Combinations of associated Legendre and hyperbolic functions . . . . . . . . . 778
7.16 Combinations of associated Legendre functions, powers, and trigonometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
7.17 A combination of an associated Legendre function and the probability integral . 781
7.18 Combinations of associated Legendre and Bessel functions . . . . . . . . . . . . 782
7.19 Combinations of associated Legendre functions and functions generated by
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
7.21 Integration of associated Legendre functions with respect to the order . . . . . 788
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 789
7.23 Combinations of Legendre polynomials and powers . . . . . . . . . . . . . . . . 791
7.24 Combinations of Legendre polynomials and other elementary functions . . . . . 792
7.25 Combinations of Legendre polynomials and Bessel functions . . . . . . . . . . . 794
7.3–7.4 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
7.31 Combinations of Gegenbauer polynomials Cν
n(x) and powers . . . . . . . . . . 795
7.32 Combinations of Gegenbauer polynomials Cν
n(x) and elementary functions . . . 797
7.325∗ Complete System of Orthogonal Step Functions . . . . . . . . . . . . . . . . . 798
7.33 Combinations of the polynomials C ν
n(x) and Bessel functions; Integration of
Gegenbauer functions with respect to the index . . . . . . . . . . . . . . . . . 798
7.34 Combinations of Chebyshev polynomials and powers . . . . . . . . . . . . . . . 800
7.35 Combinations of Chebyshev polynomials and elementary functions . . . . . . . 802
7.36 Combinations of Chebyshev polynomials and Bessel functions . . . . . . . . . . 803
7.37–7.38 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
7.39 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806
7.41–7.42 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808
7.5 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812
7.51 Combinations of hypergeometric functions and powers . . . . . . . . . . . . . . 812
7.52 Combinations of hypergeometric functions and exponentials . . . . . . . . . . . 814
7.53 Hypergeometric and trigonometric functions . . . . . . . . . . . . . . . . . . . 817
7.54 Combinations of hypergeometric and Bessel functions . . . . . . . . . . . . . . 817
7.6 Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 820
7.61 Combinations of confluent hypergeometric functions and powers . . . . . . . . 820
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials . . . . . 822
7.64 Combinations of confluent hypergeometric and trigonometric functions . . . . . 829
7.65 Combinations of confluent hypergeometric functions and Bessel functions . . . 830
xii CONTENTS
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 831
7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials,
and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834
7.68 Combinations of confluent hypergeometric functions and other special functions 839
7.69 Integration of confluent hypergeometric functions with respect to the index . . 841
7.7 Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
7.71 Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
7.72 Combinations of parabolic cylinder functions, powers, and exponentials . . . . . 842
7.73 Combinations of parabolic cylinder and hyperbolic functions . . . . . . . . . . . 843
7.74 Combinations of parabolic cylinder and trigonometric functions . . . . . . . . . 844
7.75 Combinations of parabolic cylinder and Bessel functions . . . . . . . . . . . . . 845
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849
7.77 Integration of a parabolic cylinder function with respect to the index . . . . . . 849
7.8 Meijer’s and MacRobert’s Functions (G and E) . . . . . . . . . . . . . . . . . 850
7.81 Combinations of the functions G and E and the elementary functions . . . . . 850
7.82 Combinations of the functions G and E and Bessel functions . . . . . . . . . . 854
7.83 Combinations of the functions G and E and other special functions . . . . . . . 856
8–9 Special Functions 859
8.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 859
8.11 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
8.12 Functional relations between elliptic integrals . . . . . . . . . . . . . . . . . . . 863
8.13 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865
8.14 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866
8.15 Properties of Jacobian elliptic functions and functional relationships between them 870
8.16 The Weierstrass function ℘(u) . . . . . . . . . . . . . . . . . . . . . . . . . . 873
8.17 The functions ζ(u) and σ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . 876
8.18–8.19 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
8.2 The Exponential Integral Function and Functions Generated by It . . . . . . . . 883
8.21 The exponential integral function Ei(x) . . . . . . . . . . . . . . . . . . . . . . 883
8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x . . . 886
8.23 The sine integral and the cosine integral: si x and ci x . . . . . . . . . . . . . . 886
8.24 The logarithm integral li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
8.25 The probability integral Φ(x), the Fresnel integrals S(x) and C (x), the error
function erf(x), and the complementary error function erfc(x) . . . . . . . . . 887
8.26 Lobachevskiy’s function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 891
8.3 Euler’s Integrals of the First and Second Kinds . . . . . . . . . . . . . . . . . . 892
8.31 The gamma function (Euler’s integral of the second kind): Γ(z) . . . . . . . . 892
8.32 Representation of the gamma function as series and products . . . . . . . . . . 894
8.33 Functional relations involving the gamma function . . . . . . . . . . . . . . . . 895
8.34 The logarithm of the gamma function . . . . . . . . . . . . . . . . . . . . . . . 898
8.35 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 899
8.36 The psi function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902
8.37 The function β(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906
8.38 The beta function (Euler’s integral of the first kind): B(x, y) . . . . . . . . . . 908
8.39 The incomplete beta function Bx(p, q) . . . . . . . . . . . . . . . . . . . . . . 910
8.4–8.5 Bessel Functions and Functions Associated with Them . . . . . . . . . . . . . . 910
CONTENTS xiii
8.40 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910
8.41 Integral representations of the functions Jν(z) and Nν(z) . . . . . . . . . . . . 912
8.42 Integral representations of the functions H(1)
ν (z) and H(2)
ν (z) . . . . . . . . . . 914
8.43 Integral representations of the functions Iν(z) and Kν(z) . . . . . . . . . . . . 916
8.44 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918
8.45 Asymptotic expansions of Bessel functions . . . . . . . . . . . . . . . . . . . . 920
8.46 Bessel functions of order equal to an integer plus one-half . . . . . . . . . . . . 924
8.47–8.48 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
8.49 Differential equations leading to Bessel functions . . . . . . . . . . . . . . . . . 931
8.51–8.52 Series of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933
8.53 Expansion in products of Bessel functions . . . . . . . . . . . . . . . . . . . . . 940
8.54 The zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
8.55 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
8.56 Thomson functions and their generalizations . . . . . . . . . . . . . . . . . . . 944
8.57 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945
8.58 Anger and Weber functions Jν(z) and Eν(z) . . . . . . . . . . . . . . . . . . . 948
8.59 Neumann’s and Schl¨afli’s polynomials: On(z) and S n(z) . . . . . . . . . . . . 949
8.6 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
8.60 Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
8.61 Periodic Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
8.62 Recursion relations for the coefficients A(2n)
2r , A(2n+1)
2r+1 , B(2n+1)
2r+1 , B(2n+2)
2r+2 . . . . 951
8.63 Mathieu functions with a purely imaginary argument . . . . . . . . . . . . . . . 952
8.64 Non-periodic solutions of Mathieu’s equation . . . . . . . . . . . . . . . . . . . 953
8.65 Mathieu functions for negative q . . . . . . . . . . . . . . . . . . . . . . . . . 953
8.66 Representation of Mathieu functions as series of Bessel functions . . . . . . . . 954
8.67 The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
8.7–8.8 Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 958
8.70 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958
8.71 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
8.72 Asymptotic series for large values of |ν| . . . . . . . . . . . . . . . . . . . . . . 962
8.73–8.74 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964
8.75 Special cases and particular values . . . . . . . . . . . . . . . . . . . . . . . . 968
8.76 Derivatives with respect to the order . . . . . . . . . . . . . . . . . . . . . . . 969
8.77 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
8.78 The zeros of associated Legendre functions . . . . . . . . . . . . . . . . . . . . 972
8.79 Series of associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . 972
8.81 Associated Legendre functions with integer indices . . . . . . . . . . . . . . . . 974
8.82–8.83 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975
8.84 Conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980
8.85 Toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981
8.9 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
8.90 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
8.91 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983
8.919 Series of products of Legendre and Chebyshev polynomials . . . . . . . . . . . 988
8.92 Series of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 988
8.93 Gegenbauer polynomials Cλ
n (t) . . . . . . . . . . . . . . . . . . . . . . . . . . 990
8.94 The Chebyshev polynomials Tn(x) and Un(x) . . . . . . . . . . . . . . . . . . 993
xiv CONTENTS
8.95 The Hermite polynomials H n(x) . . . . . . . . . . . . . . . . . . . . . . . . . 996
8.96 Jacobi’s polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998
8.97 The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000
9.1 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.10 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.11 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
9.12 Representation of elementary functions in terms of a hypergeometric functions . 1006
9.13 Transformation formulas and the analytic continuation of functions defined by
hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008
9.14 A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . 1010
9.15 The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . 1010
9.16 Riemann’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
9.17 Representing the solutions to certain second-order differential equations using
a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017
9.18 Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . . 1018
9.19 A hypergeometric function of several variables . . . . . . . . . . . . . . . . . . 1022
9.2 Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . 1022
9.20 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
9.21 The functions Φ(α,γ; z) and Ψ(α,γ; z) . . . . . . . . . . . . . . . . . . . . . . 1023
9.22–9.23 The Whittaker functions Mλ,μ(z) and Wλ,μ(z) . . . . . . . . . . . . . . . . . . 1024
9.24–9.25 Parabolic cylinder functions Dp(z) . . . . . . . . . . . . . . . . . . . . . . . . 1028
9.26 Confluent hypergeometric series of two variables . . . . . . . . . . . . . . . . . 1031
9.3 Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.30 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
9.31 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033
9.32 A differential equation for the G-function . . . . . . . . . . . . . . . . . . . . . 1034
9.33 Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034
9.34 Connections with other special functions . . . . . . . . . . . . . . . . . . . . . 1034
9.4 MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.41 Representation by means of multiple integrals . . . . . . . . . . . . . . . . . . 1035
9.42 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
9.5 Riemann’s Zeta Functions ζ(z,q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) 1036
9.51 Definition and integral representations . . . . . . . . . . . . . . . . . . . . . . 1036
9.52 Representation as a series or as an infinite product . . . . . . . . . . . . . . . . 1037
9.53 Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
9.54 Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038
9.55 The Lerch function Φ(z, s, v) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039
9.56 The function ξ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.6 Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . . 1040
9.61 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040
9.62 Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041
9.63 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
9.64 The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) . . . . . . . . . . 1043
9.65 Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044
9.7 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.71 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
9.72 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
CONTENTS xv
9.73 Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
9.74 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046
10 Vector Field Theory 1049
10.1–10.8 Vectors, Vector Operators, and Integral Theorems . . . . . . . . . . . . . . . . 1049
10.11 Products of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.12 Properties of scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.13 Properties of vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
10.14 Differentiation of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.21 Operators grad, div, and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
10.31 Properties of the operator ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051
10.41 Solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.51–10.61 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052
10.71–10.72 Vector integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
10.81 Integral rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . . . 1057
11 Algebraic Inequalities 1059
11.1–11.3 General Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059
11.11 Algebraic inequalities involving real numbers . . . . . . . . . . . . . . . . . . . 1059
11.21 Algebraic inequalities involving complex numbers . . . . . . . . . . . . . . . . . 1060
11.31 Inequalities for sets of complex numbers . . . . . . . . . . . . . . . . . . . . . 1061
12 Integral Inequalities 1063
12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.111 First mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.112 Second mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063
12.113 First mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . . 1063
12.114 Second mean value theorem for infinite integrals . . . . . . . . . . . . . . . . . 1064
12.21 Differentiation of Definite Integral Containing a Parameter . . . . . . . . . . . 1064
12.211 Differentiation when limits are finite . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.212 Differentiation when a limit is infinite . . . . . . . . . . . . . . . . . . . . . . . 1064
12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . . . . . . . . . . . . 1064
12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.316 Steffensen’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . 1066
12.51 Fourier Series and Related Inequalities . . . . . . . . . . . . . . . . . . . . . . 1066
12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.513 Parseval’s theorem for trigonometric Fourier series . . . . . . . . . . . . . . . . 1067
12.514 Integral representation of the nth partial sum . . . . . . . . . . . . . . . . . . . 1067
xvi CONTENTS
12.515 Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
12.516 Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
12.517 Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . . 1068
13 Matrices and Related Results 1069
13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.129 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.130 Non-negative definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.214 Positive definite and semidefinite quadratic form . . . . . . . . . . . . . . . . . 1072
13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . 1072
13.31 Differentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073
13.41 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
14 Determinants 1075
14.11 Expansion of Second- and Third-Order Determinants . . . . . . . . . . . . . . 1075
14.12 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075
14.13 Minors and Cofactors of a Determinant . . . . . . . . . . . . . . . . . . . . . . 1075
14.14 Principal Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.15* Laplace Expansion of a Determinant . . . . . . . . . . . . . . . . . . . . . . . 1076
14.16 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
14.17 Hadamard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.18 Hadamard’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.21 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
14.31 Some Special Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
CONTENTS xvii
14.311 Vandermonde’s determinant (alternant) . . . . . . . . . . . . . . . . . . . . . . 1078
14.312 Circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.313 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078
14.314 Hessian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.315 Wronskian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.316 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
14.317 Gram-Kowalewski theorem on linear dependence . . . . . . . . . . . . . . . . . 1080
15 Norms 1081
15.1–15.9 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.11 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.21 Principal Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.211 The norm ||x||1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.212 The norm ||x||2 (Euclidean or L2 norm) . . . . . . . . . . . . . . . . . . . . . 1081
15.213 The norm ||x||∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
15.31 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.311 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.312 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.313 Natural norm of unit matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.41 Principal Natural Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.411 Maximum absolute column sum norm . . . . . . . . . . . . . . . . . . . . . . . 1082
15.412 Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
15.413 Maximum absolute row sum norm . . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.51 Spectral Radius of a Square Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1083
15.511 Inequalities concerning matrix norms and the spectral radius . . . . . . . . . . 1083
15.512 Deductions from Gerschgorin’s theorem (see 15.814) . . . . . . . . . . . . . . 1083
15.61 Inequalities Involving Eigenvalues of Matrices . . . . . . . . . . . . . . . . . . . 1084
15.611 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.612 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084
15.71 Inequalities for the Characteristic Polynomial . . . . . . . . . . . . . . . . . . . 1084
15.711 Named and unnamed inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1085
15.712 Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.713 Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.714 Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.715 Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086
15.81–15.82 Named Theorems on Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.811 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.812 Sturmian separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.813 Poincare’s separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
15.814 Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.815 Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.816 Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.817 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.818 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.819 Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
15.820 Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.821 First theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
15.822 Second theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . 1089
xviii CONTENTS
15.823 Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov . . . . . . . . . . . . . . . . . . 1089
15.91 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.911 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
15.912 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091
16 Ordinary Differential Equations 1093
16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093
16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093
16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093
16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094
16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094
16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094
16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096
16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096
16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096
16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097
16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097
16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098
16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099
16.516 Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100
16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100
16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.626 Szeg¨o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101
16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102
16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
CONTENTS xix
16.712 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.81–16.82 Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.811 Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1103
16.822 Comparison theorem for non-oscillation . . . . . . . . . . . . . . . . . . . . . . 1104
16.823 Necessary and sufficient conditions for non-oscillation . . . . . . . . . . . . . . 1104
16.91 Some Growth Estimates for Solutions of Second-Order Equations . . . . . . . . 1104
16.911 Strictly increasing and decreasing solutions . . . . . . . . . . . . . . . . . . . . 1104
16.912 General result on dominant and subdominant solutions . . . . . . . . . . . . . 1104
16.913 Estimate of dominant solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.914 A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
16.92 Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.921 All solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.922 If all solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.923 If a(x) → ∞ monotonically as x → ∞, then all solutions of . . . . . . . . . . . 1106
16.924 Consider the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
16.93 Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106
17 Fourier, Laplace, and Mellin Transforms 1107
17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107
17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . . . . . . . . 1107
17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . 1108
17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117
17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . . 1118
17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
17.24 Table of Fourier transform pairs for spherically symmetric functions . . . . . . . 1120
17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1121
17.32 Basic properties of the Fourier sine and cosine transforms . . . . . . . . . . . . 1121
17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 1122
17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1126
17.35 Relationships between transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129
17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . . . . . . . . 1130
17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131
18 The z-Transform 1135
18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms . . . . . . . . . . . . . . . . . 1135
18.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
18.2 Bilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136
18.3 Unilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
References 1141
Supplemental references 1145
Index of Functions and Constants 1151
General Index of Concep
|
| Côte titre : |
Fs/12411,Fs/10427 |
|  |
Table of integrals, series and products
