Table of integrals, series, and products:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Academic Press
2007
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| Ausgabe: | 7. ed. |
| Schlagworte: | |
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| Beschreibung: | Auch als CD-ROM-Ausg. u.d.T.: Table of integrals, series, and products. - Aus dem Russ. übers. |
| Beschreibung: | XLV, 1171 S. 1 CD-ROM (12 cm) |
| ISBN: | 0123736374 9780123736376 |
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| 100 | 1 | |a Gradštejn, Izrailʹ S. |d 1899-1958 |e Verfasser |0 (DE-588)11526194X |4 aut | |
| 240 | 1 | 0 | |a Tablicy integralov, summ, rjadov, i proizvedenij |
| 245 | 1 | 0 | |a Table of integrals, series, and products |c I. S. Gradshteyn and I. M. Ryzhik. Alan Jeffrey, ed. ... |
| 250 | |a 7. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Academic Press |c 2007 | |
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Datensatz im Suchindex
| _version_ | 1804135459072245760 |
|---|---|
| adam_text | New to This Edition
The seventh edition includes a fully searchable CD version of the book.This world-renowned
guide has been thoroughly updated with corrections received since the publication of the sixth
edition in
2000,
together with a considerable amount of new material acquired from isolated sources.
The comprehensive coverage of integrals and of many special functions that arise in all
aspects of mathematics and its applications make the book an essential reference work for the
specialist, and also for the general user whose requirements are less specialized but frequently
involve a greater diversity of integrals.
The integrals are very useful, but this book includes many other features that will be helpful
to the reader, especially graduate students.The sections on Hermite and Legendre
polynomials are especially helpful for students of Electricity and Magnetism, Quantum Mechanics,
and Mathematical physics (they won t have to hunt in several books to find what they need).
Barry Simon, California Institute of Technology
This book is to the
CRC
Mathematical Tables as the unabridged Oxford English Dictionary is to
Webster s Collegiate. Besides being big, it s easy to find things in, because of the way the
integrals are organized into classes...lt really helped me through
grad
school.
Phil Hobbs.Amazon Review
Contents
Preface
to the Seventh Edition
xxi
Acknowledgments
xxiii
The Order of Presentation of the Formulas
xxvii
use of the Tables
xxxi
Index of Special Functions
xxxix
Notation
xliii
Note on the Bibliographic References
xlvii
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
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
П(аг)
................... 51
1.49
The hyperbolic amplitude (the Gudermannian) gd
χ
............... 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
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
lia;
...................... 74
2.15
Forms containing pairs of binomials: a + bx and a
+
βχ
............. 78
2.16
Forms containing the trinomial a + bxk +cx2k
.................. 78
2.17
Forms containing the quadratic trinomial a + bx
+
ex2 and powers of a;
.... 79
2.18
Forms containing the quadratic trinomial a + bx
+
ex2 and the binomial a
+
βχ
81
2.2
Algebraic Functions
................................ 82
2.20
Introduction
.................................... 82
2.21
Forms containing the binomial a
+
bxk and y/x
................. 83
CONTENTS
2.22-2.23
Forms containing
ý
{a
+
Ъх)к
........................... 84
2.24
Forms containing
л/
a + bx and the binomial a
+
βχ
............... 88
2.25
Forms containing /a
+
bx
+
ex2
......................... 92
2.26
Forms containing y/a
+
bx
+
ex2 and integral powers of
ж
............ 94
2.27
Forms containing
л/а
+
ex2 and integral powers of
ж
............... 99
2.28
Forms containing /a
+
bx
+
ex2 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
ж
.............. 106
2.4
Hyperbolic Functions
................................ 110
2.41-2.43
Powers of sinha;, cosha;, tanha;, and cothx
................... 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 func¬
tions of the argument
............................... 161
2.55-2.56
Rational functions of the sine and cosine
..................... 171
2.57
Integrals containing /a±bsmx 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 a;
........ 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
vüi 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
χ
and powers of binomials of the form (a
+
βχ)
. . 315
3.24-3.27
Powers of x, of binomials of the form a +
βχρ
and of polynomials in a;
..... 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
χ
and of trigonometric functions
.............. 447
3.82-3.83
Powers of trigonometric functions combined with other powers
......... 459
3.84
Integrals containing
л
/í
—
k2 sin2 x,
л/1
—
к2
cos2
χ,
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 expo¬
nentials
....................................... 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
lna;
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
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
Besse! Functions
.................................. 629
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(χ)
................................. 658
6.5-6.7
Bessel Functions
.................................. 659
6.51
Bessel functions
.................................. 659
6.52
Bessel functions combined with
χ
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 argu¬
ments 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
Lömmel
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 C^{x) and powers
.......... 795
7.32
Combinations of
Gegenbauer
polynomials C^(ar) and elementary functions
. . . 797
7.325*
Complete System of Orthogonal Step Functions
................. 798
7.33
Combinations of the polynomials
С^{х)
and Bessei 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
CONTENTS
7.66
Combinations of confluent hypergeometric functions, Bessel functions, and powers
831
7.67
Combinations of confluent hypergeometric functions, Bessel functions, expo¬
nentials, 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 p(u)
.......................... 873
8.17
The functions C(u) and a(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
Еі(ж)
...................... 883
8.22
The hyperbolic sine integral shi
χ
and the hyperbolic cosine integral
chia;
. . . 886
8.23
The sine integral and the cosine integral:
sia;
and
сіж
.............. 886
8.24
The logarithm integral
1і(ж)
............................ 887
8.25
The probability integral
Ф(х),
the Fresnel integrals S(x) and
С(х),
the error
function erf(x), and the complementary error function erfc(a;)
......... 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): T(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(χ)
............................... 902
8.37
The function
β(χ)
................................. 906
8.38
The beta function (Euler s integral of the first kind): B(x,y)
.......... 908
8.39
The incomplete beta function
Вж(р,
q)......................
910
8.4-8.5
Bessel Functions and Functions Associated with Them
.............. 910
CONTENTS
8.40
Definitions
..................................... 910
8.41 Integral
representations of the functions Jv{z) and Nu(z)
............ 912
8.42
Integral representations of the functions H^
(z)
and
Н^
(z)
.......... 914
8.43
Integral representations of the functions Iv(z) and
К
v{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
Lömmel
functions
................................. 945
8.58
Anger and Weber functions
Λν(ζ)
and Ev(z)
................... 948
8.59
Neumann s and
Schläfli s
polynomials: On(z) and Sn(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^f1
,
A^1},
В^1]
,
B^+ţ2)
.... 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 u
...................... 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 C^(t)
.......................... 990
8.94
The Chebyshev polynomials Tn(x) and Un(x)
.................. 993
xiv
CONTENTS
8.95
The Hermite
polynomials
Нп(х)
......................... 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,
7;
ζ)
and ^f(a,j;z)
...................... 1023
9.22-9.23
The Whittaker functions
Μλ,μ(ζ)
and Wx^(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
................................
Ю34
9.34
Connections with other special functions
..................... 1034
9.4
MacRobert s iJ-Function
..............................
Ю35
9.41
Representation by means of multiple integrals
.................. 1035
9.42
Functional relations
................................ 1035
9-5 Riemann s
Zeta
Functions C(z,q) and
ζ(ζ),
and the Functions
Φ(ζ,8,ν)
and
ξ(β)
1036
9.51
Definition and integral representations
...................... 1036
9.52
Representation as a series or as an infinite product
................ 1037
9.53
Functional relations
................................
Ю37
9.54
Singular points and zeros
.............................
Ю38
9.55
The Lerch function
Φ(ζ,
s,
υ)...........................
1039
9.56
The function
ξ
(s)
................................. 1040
9.6
Bernoulli Numbers and Polynomials,
Euler
Numbers
............... 1040
9.61
Bernoulli numbers
.................................
Ю40
9.62
Bernoulli polynomials
...............................
Ю41
9.63
Euler
numbers
...................................
IO43
9.64
The functions v{x), v{x,
α), μ(χ,β), μ(χ,β,α),
and X(x,y)
.......... 1043
9.65
Euler
polynomials
.................................
IO44
9.7
Constants
......................................
IO45
9.71
Bernoulli numbers
.................................
IO45
9.72
Euler
numbers
..................................
IO45
CONTENTS
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 V
........................... 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
Holder 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
Parsevai 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
...................................
Ю77
14.31
Some Special Determinants
............................ 1078
CONTENTS
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
ЦхЦх
.................................. 1081
15.212
The norm
||х|[2
(Euclidean or L2 norm)
..................... 1081
15.213
The norm
ІІхЩ
.................................. 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 Szegö 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.....................................
H°3
CONTENTS
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)
—>
oo monotonically as
χ
—>
co,
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 Concepts
1161
|
| adam_txt |
New to This Edition
The seventh edition includes a fully searchable CD version of the book.This world-renowned
guide has been thoroughly updated with corrections received since the publication of the sixth
edition in
2000,
together with a considerable amount of new material acquired from isolated sources.
The comprehensive coverage of integrals and of many special functions that arise in all
aspects of mathematics and its applications make the book an essential reference work for the
specialist, and also for the general user whose requirements are less specialized but frequently
involve a greater diversity of integrals.
"The integrals are very useful, but this book includes many other features that will be helpful
to the reader, especially graduate students.The sections on Hermite and Legendre
polynomials are especially helpful for students of Electricity and Magnetism, Quantum Mechanics,
and Mathematical physics (they won't have to hunt in several books to find what they need)."
Barry Simon, California Institute of Technology
"This book is to the
CRC
Mathematical Tables as the unabridged Oxford English Dictionary is to
Webster's Collegiate. Besides being big, it's easy to find things in, because of the way the
integrals are organized into classes.lt really helped me through
grad
school."
Phil Hobbs.Amazon Review
Contents
Preface
to the Seventh Edition
xxi
Acknowledgments
xxiii
The Order of Presentation of the Formulas
xxvii
use of the Tables
xxxi
Index of Special Functions
xxxix
Notation
xliii
Note on the Bibliographic References
xlvii
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
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"
П(аг)
. 51
1.49
The hyperbolic amplitude (the Gudermannian) gd
χ
. 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
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
lia;"
. 74
2.15
Forms containing pairs of binomials: a + bx and a
+
βχ
. 78
2.16
Forms containing the trinomial a + bxk +cx2k
. 78
2.17
Forms containing the quadratic trinomial a + bx
+
ex2 and powers of a;
. 79
2.18
Forms containing the quadratic trinomial a + bx
+
ex2 and the binomial a
+
βχ
81
2.2
Algebraic Functions
. 82
2.20
Introduction
. 82
2.21
Forms containing the binomial a
+
bxk and y/x
. 83
CONTENTS
2.22-2.23
Forms containing
ý
{a
+
Ъх)к
. 84
2.24
Forms containing
л/
a + bx and the binomial a
+
βχ
. 88
2.25
Forms containing \/a
+
bx
+
ex2
. 92
2.26
Forms containing y/a
+
bx
+
ex2 and integral powers of
ж
. 94
2.27
Forms containing
л/а
+
ex2 and integral powers of
ж
. 99
2.28
Forms containing \/a
+
bx
+
ex2 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
ж
. 106
2.4
Hyperbolic Functions
. 110
2.41-2.43
Powers of sinha;, cosha;, tanha;, and cothx
. 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 func¬
tions of the argument
. 161
2.55-2.56
Rational functions of the sine and cosine
. 171
2.57
Integrals containing \/a±bsmx 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 a;
. 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
vüi 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
χ
and powers of binomials of the form (a
+
βχ)
. . 315
3.24-3.27
Powers of x, of binomials of the form a +
βχρ
and of polynomials in a;
. 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
χ
and of trigonometric functions
. 447
3.82-3.83
Powers of trigonometric functions combined with other powers
. 459
3.84
Integrals containing
л
/í
—
k2 sin2 x,
л/1
—
к2
cos2
χ,
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 expo¬
nentials
. 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
lna;
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
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
Besse! Functions
. 629
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(χ)
. 658
6.5-6.7
Bessel Functions
. 659
6.51
Bessel functions
. 659
6.52
Bessel functions combined with
χ
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 argu¬
ments 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
Lömmel
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 C^{x) and powers
. 795
7.32
Combinations of
Gegenbauer
polynomials C^(ar) and elementary functions
. . . 797
7.325*
Complete System of Orthogonal Step Functions
. 798
7.33
Combinations of the polynomials
С^{х)
and Bessei 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
CONTENTS
7.66
Combinations of confluent hypergeometric functions, Bessel functions, and powers
831
7.67
Combinations of confluent hypergeometric functions, Bessel functions, expo¬
nentials, 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 p(u)
. 873
8.17
The functions C(u) and a(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
Еі(ж)
. 883
8.22
The hyperbolic sine integral shi
χ
and the hyperbolic cosine integral
chia;
. . . 886
8.23
The sine integral and the cosine integral:
sia;
and
сіж
. 886
8.24
The logarithm integral
1і(ж)
. 887
8.25
The probability integral
Ф(х),
the Fresnel integrals S(x) and
С(х),
the error
function erf(x), and the complementary error function erfc(a;)
. 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): T(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(χ)
. 902
8.37
The function
β(χ)
. 906
8.38
The beta function (Euler's integral of the first kind): B(x,y)
. 908
8.39
The incomplete beta function
Вж(р,
q).
910
8.4-8.5
Bessel Functions and Functions Associated with Them
. 910
CONTENTS
8.40
Definitions
. 910
8.41 Integral
representations of the functions Jv{z) and Nu(z)
. 912
8.42
Integral representations of the functions H^
(z)
and
Н^
(z)
. 914
8.43
Integral representations of the functions Iv(z) and
К
v{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
Lömmel
functions
. 945
8.58
Anger and Weber functions
Λν(ζ)
and Ev(z)
. 948
8.59
Neumann's and
Schläfli's
polynomials: On(z) and Sn(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^f1
,
A^1},
В^1]
,
B^+ţ2)
. 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 \u\
. 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 C^(t)
. 990
8.94
The Chebyshev polynomials Tn(x) and Un(x)
. 993
xiv
CONTENTS
8.95
The Hermite
polynomials
Нп(х)
. 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,
7;
ζ)
and ^f(a,j;z)
. 1023
9.22-9.23
The Whittaker functions
Μλ,μ(ζ)
and Wx^(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
.
Ю34
9.34
Connections with other special functions
. 1034
9.4
MacRobert's iJ-Function
.
Ю35
9.41
Representation by means of multiple integrals
. 1035
9.42
Functional relations
. 1035
9-5 Riemann's
Zeta
Functions C(z,q) and
ζ(ζ),
and the Functions
Φ(ζ,8,ν)
and
ξ(β)
1036
9.51
Definition and integral representations
. 1036
9.52
Representation as a series or as an infinite product
. 1037
9.53
Functional relations
.
Ю37
9.54
Singular points and zeros
.
Ю38
9.55
The Lerch function
Φ(ζ,
s,
υ).
1039
9.56
The function
ξ
(s)
. 1040
9.6
Bernoulli Numbers and Polynomials,
Euler
Numbers
. 1040
9.61
Bernoulli numbers
.
Ю40
9.62
Bernoulli polynomials
.
Ю41
9.63
Euler
numbers
.
IO43
9.64
The functions v{x), v{x,
α), μ(χ,β), μ(χ,β,α),
and X(x,y)
. 1043
9.65
Euler
polynomials
.
IO44
9.7
Constants
.
IO45
9.71
Bernoulli numbers
.
IO45
9.72
Euler
numbers
.
IO45
CONTENTS
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 V
. 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
Holder'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
Parsevai'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
.
Ю77
14.31
Some Special Determinants
. 1078
CONTENTS
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
ЦхЦх
. 1081
15.212
The norm
||х|[2
(Euclidean or L2 norm)
. 1081
15.213
The norm
ІІхЩ
. 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 Szegö'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.
H°3
CONTENTS
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)
—>
oo monotonically as
χ
—>
co,
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 Concepts
1161 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gradštejn, Izrailʹ S. 1899-1958 Ryžik, Iosif M. |
| author_GND | (DE-588)11526194X |
| author_facet | Gradštejn, Izrailʹ S. 1899-1958 Ryžik, Iosif M. |
| author_role | aut aut |
| author_sort | Gradštejn, Izrailʹ S. 1899-1958 |
| author_variant | i s g is isg i m r im imr |
| building | Verbundindex |
| bvnumber | BV021652062 |
| classification_rvk | QH 100 SH 500 |
| classification_tum | MAT 001k MAT 330k MAT 260k |
| ctrlnum | (OCoLC)315684950 (DE-599)BVBBV021652062 |
| dewey-full | 515.0212 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.0212 |
| dewey-search | 515.0212 |
| dewey-sort | 3515.0212 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | 7. ed. |
| format | Book |
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| genre | (DE-588)4155008-0 Formelsammlung gnd-content (DE-588)4184303-4 Tabelle gnd-content |
| genre_facet | Formelsammlung Tabelle |
| id | DE-604.BV021652062 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:03:21Z |
| indexdate | 2024-07-09T20:40:50Z |
| institution | BVB |
| isbn | 0123736374 9780123736376 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014866700 |
| oclc_num | 315684950 |
| open_access_boolean | |
| owner | DE-945 DE-384 DE-29T DE-92 DE-703 DE-19 DE-BY-UBM DE-20 DE-739 DE-91G DE-BY-TUM DE-83 DE-355 DE-BY-UBR |
| owner_facet | DE-945 DE-384 DE-29T DE-92 DE-703 DE-19 DE-BY-UBM DE-20 DE-739 DE-91G DE-BY-TUM DE-83 DE-355 DE-BY-UBR |
| physical | XLV, 1171 S. 1 CD-ROM (12 cm) |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Academic Press |
| record_format | marc |
| spelling | Gradštejn, Izrailʹ S. 1899-1958 Verfasser (DE-588)11526194X aut Tablicy integralov, summ, rjadov, i proizvedenij Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik. Alan Jeffrey, ed. ... 7. ed. Amsterdam [u.a.] Academic Press 2007 XLV, 1171 S. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Auch als CD-ROM-Ausg. u.d.T.: Table of integrals, series, and products. - Aus dem Russ. übers. Mathematics - Tables blmsh Mathematik Reihe (DE-588)4049197-3 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Summe (DE-588)4193845-8 gnd rswk-swf Integral (DE-588)4131477-3 gnd rswk-swf Produkt (DE-588)4139399-5 gnd rswk-swf Reihe Musik (DE-588)4368335-6 gnd rswk-swf Produkt Mathematik (DE-588)4126359-5 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content (DE-588)4184303-4 Tabelle gnd-content Produkt (DE-588)4139399-5 s DE-604 Integral (DE-588)4131477-3 s Summe (DE-588)4193845-8 s Reihe (DE-588)4049197-3 s Produkt Mathematik (DE-588)4126359-5 s 1\p DE-604 Mathematik (DE-588)4037944-9 s 2\p DE-604 Reihe Musik (DE-588)4368335-6 s 3\p DE-604 Ryžik, Iosif M. Verfasser aut Jeffrey, Alan Sonstige oth Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014866700&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014866700&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gradštejn, Izrailʹ S. 1899-1958 Ryžik, Iosif M. Table of integrals, series, and products Mathematics - Tables blmsh Mathematik Reihe (DE-588)4049197-3 gnd Mathematik (DE-588)4037944-9 gnd Summe (DE-588)4193845-8 gnd Integral (DE-588)4131477-3 gnd Produkt (DE-588)4139399-5 gnd Reihe Musik (DE-588)4368335-6 gnd Produkt Mathematik (DE-588)4126359-5 gnd |
| subject_GND | (DE-588)4049197-3 (DE-588)4037944-9 (DE-588)4193845-8 (DE-588)4131477-3 (DE-588)4139399-5 (DE-588)4368335-6 (DE-588)4126359-5 (DE-588)4155008-0 (DE-588)4184303-4 |
| title | Table of integrals, series, and products |
| title_alt | Tablicy integralov, summ, rjadov, i proizvedenij |
| title_auth | Table of integrals, series, and products |
| title_exact_search | Table of integrals, series, and products |
| title_exact_search_txtP | Table of integrals, series, and products |
| title_full | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik. Alan Jeffrey, ed. ... |
| title_fullStr | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik. Alan Jeffrey, ed. ... |
| title_full_unstemmed | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik. Alan Jeffrey, ed. ... |
| title_short | Table of integrals, series, and products |
| title_sort | table of integrals series and products |
| topic | Mathematics - Tables blmsh Mathematik Reihe (DE-588)4049197-3 gnd Mathematik (DE-588)4037944-9 gnd Summe (DE-588)4193845-8 gnd Integral (DE-588)4131477-3 gnd Produkt (DE-588)4139399-5 gnd Reihe Musik (DE-588)4368335-6 gnd Produkt Mathematik (DE-588)4126359-5 gnd |
| topic_facet | Mathematics - Tables Mathematik Reihe Summe Integral Produkt Reihe Musik Produkt Mathematik Formelsammlung Tabelle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014866700&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014866700&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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