Table of integrals, series, and products:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
San Diego [u.a.]
Acad. Press
2000
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| Edition: | 6. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auch als CD-ROM-Ausg. u.d.T.: Table of integrals, series, and products. - Aus dem Russ. übers. |
| Physical Description: | XLVII, 1163 S. |
| ISBN: | 0122947576 |
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| 245 | 1 | 0 | |a Table of integrals, series, and products |c I. S. Gradshteyn and I. M. Ryzhik |
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| _version_ | 1804128154838630400 |
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| adam_text | Titel: Table of integrals, series, and products
Autor: Gradštejn, Izrailʹ S
Jahr: 2000
Contents
Preface to the Sixth Edition xxi
Acknowledgments xxiii
The order of presentation of the formulas xxvii
Use of the tables xxxi
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 2
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 G
0.21 The convergence of numerical series G
0.22 Convergence tests G
0.23-0.24 Examples of numerical series 8
0.25 Infinite products 14
0.2G Examples of infinite products 14
0.3 Functional series 15
0.30 Definitions and theorems 15
0.31 Power series 1G
0.32 Fourier series 18
0.33 Asymptotic series 20
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) 21
0.43 The nth derivative of a composite function 21
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 2G
1.2 The Exponential Function 26
v
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 27
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) 30
1.33 The representation of trigonometric and hyperbolic functions of multiples of
the argument (angle) in terms of powers of these functions 32
1.34 Certain sums of trigonometric and hyperbolic functions 35
1.35 Sums of powers of trigonometric functions of multiple angles 36
1.36 Sums of products of trigonometric functions of multiple angles 37
1.37 Sums of tangents of multiple angles 38
1.38 Sums leading to hyperbolic tangents and cotangents 38
1.39 The representation of cosines and sines of multiples of the angle as finite products 39
1.41 The expansion of trigonometric and hyperbolic functions in power series .... 41
1.42 Expansion in series of simple fractions 42
1.43 Representation in the form of an infinite product 43
1.44-1.45 Trigonometric (Fourier) series 44
1.46 Series of products of exponential and trigonometric functions 48
1.47 Series of hyperbolic functions 49
1.48 Lobachevskiy s Angle of parallelism 11(3;) 49
1.49 The hyperbolic amplitude (the Gudermannian) gdx 50
1.5 The Logarithm 51
1.51 Series representation 51
1.52 Series of logarithms (cf. 1.431) 53
1.6 The Inverse Trigonometric and Hyperbolic Functions 54
1.61 The domain of definition 54
1.62-1.63 Functional relations 54
1.64 Series representations 58
2 Indefinite Integrals of Elementary Functions 61
2.0 Introduction 61
2.00 General remarks 61
2.01 The basic integrals 61
2.02 General formulas 62
2.1 Rational functions 64
2.10 General integration rules 64
2.11-2.13 Forms containing the binomial a + bxk 66
2.14 Forms containing the binomial 1 ± xn 72
2.15 Forms containing pairs of binomials: a + bx and a + [3x 76
2.16 Forms containing the trinomial a + bxk + cx2k . 76
2-17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x .... 77
2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial a + (3x 79
2.2 Algebraic functions 80
2.20 Introduction 80
2.21 Forms containing the binomial a + bxk and */x ... 81
CONTENTS v[i
2.22-2.23 Forms containing y/{a + bx)k 83
2.24 Forms containing y/a + bx and the binomial a + 0x 8G
2.25 Forms containing y/a + bx + cx2 90
2.26 Forms containing y/a + b + cx2 and integral powers of x 92
2.27 Forms containing y/a + cx2 and integral powers of x 97
2.28 Forms containing y/a+ bx + cx2 and first-and second-degree polynomials . . . 101
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals 102
2.3 The Exponential Function 104
2.31 Forms containing eax 104
2.32 The exponential combined with rational functions of x 104
2.4 Hyperbolic Functions 105
2.41-2.43 Powers of sinhx, coshx, tanhx, and cothx 105
2.44-2.45 Rational functions of hyperbolic functions 121
2.4G Algebraic functions of hyperbolic functions 128
2.47 Combinations of hyperbolic functions and powers 136
2.48 Combinations of hyperbolic functions, exponentials, and powers 145
2.5-2.G Trigonometric Functions 147
2.50 Introduction 147
2.51-2.52 Powers of trigonometric functions 147
2.53-2.54 Sines and cosines of multiple angles and of linear and more complicated func¬
tions of the argument 157
2.55-2.56 Rational functions of the sine and cosine . . . 167
2.57 Integrals containing y/a ± bsinx or y/a ± bcosx 175
2.58-2.62 Integrals reducible to elliptic and pseudo-elliptic integrals 180
2.63-2.65 Products of trigonometric functions and powers 210
2.66 Combinations of trigonometric functions and exponentials 222
2.67 Combinations of trigonometric and hyperbolic functions 227
2.7 Logarithms and Inverse-Hyperbolic Functions 233
2.71 The logarithm 233
2.72-2.73 Combinations of logarithms and algebraic functions 233
2.74 Inverse hyperbolic functions 236
2.8 Inverse Trigonometric Functions 237
2.81 Arcsines and arccosines 237
2.82 The arcsecant, the arccosecant, the arctangent and the arccotangent 238
2.83 Combinations of arcsine or arccosine and algebraic functions 238
2.84 Combinations of the arcsecant and arccosecant with powers of x 240
2.85 Combinations of the arctangent and arccotangent with algebraic functions . . . 240
3-4 Definite Integrals of Elementary Functions 243
3.0 Introduction 243
3.01 Theorems of a general nature 243
3.02 Change of variable in a definite integral 244
3.03 General formulas 245
3.04 Improper integrals 247
3.05 The principal values of improper integrals 248
3.1-3.2 Power and Algebraic Functions 248
3.11 Rational functions 249
viii
CONTENTS
3.12 Products of rational functions and expressions that can be reduced to square
roots of first-and second-degree polynomials 249
3.13-3.17 Expressions that can be reduced to square roots of third-and fourth-degree
polynomials and their products with rational functions 250
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials
and their products with rational functions 310
3.19-3.23 Combinations of powers of x and powers of binomials of the form (a + 0x) . . 312
3.24-3.27 Powers of x, of binomials of the form a + /3xp and of polynomials in x 319
3.3-3.4 Exponential Functions 331
3.31 Exponential functions 331
3.32-3.34 Exponentials of more complicated arguments 333
3.35 Combinations of exponentials and rational functions 336
3.3G-3.37 Combinations of exponentials and algebraic functions 340
3.38-3.39 Combinations of exponentials and arbitrary powers 342
3.41-3.44 Combinations of rational functions of powers and exponentials 349
3.45 Combinations of powers and algebraic functions of exponentials 358
3.46-3.48 Combinations of exponentials of more complicated arguments and powers . . . 360
3.5 Hyperbolic Functions 365
3.51 Hyperbolic functions 366
3.52-3.53 Combinations of hyperbolic functions and algebraic functions 369
3.54 Combinations of hyperbolic functions and exponentials 376
3.55-3.56 Combinations of hyperbolic functions, exponentials, and powers 380
3.6-4.1 Trigonometric Functions 384
3.61 Rational functions of sines and cosines and trigonometric functions of multiple
angles 385
3.62 Powers of trigonometric functions 388
3.63 Powers of trigonometric functions and trigonometric functions of linear functions 390
3.64-3.65 Powers and rational functions of trigonometric functions 395
3.66 Forms containing powers of linear functions of trigonometric functions 399
3.67 Square roots of expressions containing trigonometric functions 402
3.68 Various forms of powers of trigonometric functions 404
3.69-3.71 Trigonometric functions of more complicated arguments 408
3.72-3.74 Combinations of trigonometric and rational functions 417
3.75 Combinations of trigonometric and algebraic functions 428
3.76-3.77 Combinations of trigonometric functions and powers 429
3.78-3.81 Rational functions of x and of trigonometric functions 440
3.82-3.83 Powers of trigonometric functions combined with other powers 453
3.84 Integrals containing /l - k2 sin2 x, Vf - k2 cos2 a:, and similar expressions . . 466
3.85-3.88 Trigonometric functions of more complicated arguments combined with powers 469
3.89-3.91 Trigonometric functions and exponentials 479
3.92 Trigonometric functions of more complicated arguments combined with expo¬
nentials 487
3.93 Trigonometric and exponential functions of trigonometric functions 490
3.94-3.97 Combinations involving trigonometric functions, exponentials, and powers ... 492
3.98 -3.99 Combinations of trigonometric and hyperbolic functions 504
4.11-4.12 Combinations involving trigonometric and hyperbolic functions and powers ... 511
4.13 Combinations of trigonometric and hyperbolic functions and exponentials . ... 517
CONTENTS ix
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 520
4.2-4.4 Logarithmic Functions 522
4.21 Logarithmic functions 522
4.22 Logarithms of more complicated arguments 525
4.23 Combinations of logarithms and rational functions 530
4.24 Combinations of logarithms and algebraic functions 532
4.25 Combinations of logarithms and powers 534
4.26-4.27 Combinations involving powers of the logarithm and other powers 537
4.28 Combinations of rational functions of In# and powers 549
4.29 -4.32 Combinations of logarithmic functions of more complicated arguments and powers 551
4.33-4.34 Combinations of logarithms and exponentials 567
4.35 4.36 Combinations of logarithms, exponentials, and powers 568
4.37 Combinations of logarithms and hyperbolic functions 574
4.38-4.41 Logarithms and trigonometric functions 577
4.42-4.43 Combinations of logarithms, trigonometric functions, and powers 590
4.44 Combinations of logarithms, trigonometric functions, and exponentials 595
4.5 Inverse Trigonometric Functions 596
4.51 Inverse trigonometric functions 596
4.52 Combinations of arcsines, arccosines, and powers 596
4.53- 4.54 Combinations of arctangents, arccotangents, and powers 597
4.55 Combinations of inverse trigonometric functions and exponentials 601
4.56 A combination of the arctangent and a hyperbolic function G01
4.57 Combinations of inverse and direct trigonometric functions 601
4.58 A combination involving an inverse and a direct trigonometric function and a
power 503
4.59 Combinations of inverse trigonometric functions and logarithms 603
4.6 Multiple Integrals 604
4.60 Change of variables in multiple integrals 604
4.61 Change of the order of integration and change of variables 604
4.62 Double and triple integrals with constant limits 607
4.63-4.64 Multiple integrals 609
5 Indefinite Integrals of Special Functions 615
5.1 Elliptic Integrals and Functions 615
5.11 Complete elliptic integrals 615
5.12 Elliptic integrals 616
5.13 Jacobian elliptic functions 618
5.14 Weierstrass elliptic functions 622
5.2 The Exponential Integral Function G22
5.21 The exponential Integral function 622
5.22 Combinations of the exponential integral function and powers 622
5.23 Combinations of the exponential integral and the exponential 622
5.3 The Sine Integral and the Cosine Integral 623
5.4 The Probability Integral and Fresnel Integrals G23
5.5 Bessel Functions 624
X
CONTENTS
6-7 Definite Integrals of Special Functions ^25
6.1 Elliptic Integrals and Functions ^25
6.11 Forms containing F(x, k) G2^
6.12 Forms containing E(x, k) G26
6.13 Integration of elliptic integrals with respect to the modulus 626
6.14-6.15 Complete elliptic integrals C26
6.16 The theta function 627
6.17 Generalized elliptic integrals 628
6.2-6.3 The Exponential Integral Function and Functions Generated by It 630
6.21 The logarithm integral 630
6.22-6.23 The exponential integral function 631
6.24-6.26 The sine integral and cosine integral functions 633
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 638
6.28-6.31 The probability integral 638
6.32 Fresnel integrals 642
6.4 The Gamma Function and Functions Generated by It 644
6.41 The gamma function 644
6.42 Combinations of the gamma function, the exponential, and powers 645
6.43 Combinations of the gamma function and trigonometric functions 648
6.44 The logarithm of the gamma function* 649
6.45 The incomplete gamma function 650
6.46-6.47 The function ip{x) 651
6.5-6.7 Bessel Functions 652
6.51 Bessel functions 653
6.52 Bessel functions combined with x and x2 657
6.53-6.54 Combinations of Bessel functions and rational functions 662
6.55 Combinations of Bessel functions and algebraic functions 666
6.56-6.58 Combinations of Bessel functions and powers 667
6.59 Combinations of powers and Bessel functions of more complicated arguments . 681
6.61 Combinations of Bessel functions and exponentials 686
6.62-6.63 Combinations of Bessel functions, exponentials, and powers 691
6.64 Combinations of Bessel functions of more complicated arguments, exponentials,
and powers 701
6.65 Combinations of Bessel and exponential functions of more complicated argu¬
ments and powers 703
6.66 Combinations of Bessel, hyperbolic, and exponential functions 705
6.67-6.68 Combinations of Bessel and trigonometric functions 709
6.69-6.74 Combinations of Bessel and trigonometric functions and powers 719
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers . 735
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions 739
6.77 Combinations of Bessel functions and the logarithm, or arctangent 739
6.78 Combinations of Bessel and other special functions 740
6.79 Integration of Bessel functions with respect to the order 741
6.8 Functions Generated by Bessel Functions 745
6.81 Struve functions 745
6.82 Combinations of Struve functions, exponentials, and powers 747
6.83 Combinations of Struve and trigonometric functions 748
xl
748
752
754
755
755
750
759
759
702
702
703
709
771
772
774
774
780
781
782
784
785
787
788
788
790
791
793
794
795
790
800
801
80G
800
807
810
810
814
814
815
822
824
824
Combinations of Struve and Bessel functions
Lommel functions
Thomson functions
Mathieu Functions
Mathieu functions
Combinations of Mathieu, hyperbolic, and trigonometric functions
Combinations of Mathieu and Bessel functions
Relationships between eigenfunctions of the Helmholtz equation in different
coordinate systems
Associated Legendre Functions
Associated Legendre functions
Combinations of associated Legendre functions and powers
Combinations of associated Legendre functions, exponentials, and powers . . .
Combinations of associated Legendre and hyperbolic functions
Combinations of associated Legendre functions, powers, and trigonometric
functions
A combination of an associated Legendre function and the probability integral .
Combinations of associated Legendre and Bessel functions
Combinations of associated Legendre functions and functions generated by
Bessel functions
Integration of associated Legendre functions with respect to the order
Combinations of Legendre polynomials, rational functions, and algebraic functions
Combinations of Legendre polynomials and powers
Combinations of Legendre polynomials and other elementary functions
Combinations of Legendre polynomials and Bessel functions
Orthogonal Polynomials
Combinations of Gegenbauer polynomials C„(x) and powers
Combinations of Gegenbauer polynomials C„(x) and some elementary functions
Combinations of the polynomials C^ix) and Bessel functions. Integration of
Gegenbauer functions with respect to the index
Combinations of Chebyshev polynomials and powers
Combinations of Chebyshev polynomials and some elementary functions ....
Combinations of Chebyshev polynomials and Bessel functions
Hermite polynomials
Jacobi polynomials
Laguerre polynomials
Hypergeometric Functions
Combinations of hypergeometric functions and powers
Combinations of hypergeometric functions and exponentials
Hypergeometric and trigonometric functions
Combinations of hypergeometric and Bessel functions
Confluent Hypergeometric Functions
Combinations of confluent hypergeometric functions and powers
Combinations of confluent hypergeometric functions and exponentials
Combinations of confluent hypergeometric and trigonometric functions
Combinations of confluent hypergeometric functions and Bessel functions . . .
Combinations of confluent hypergeometric functions, Bessel functions, and powers
CONTENTS
7.67 Combinations of confluent hypergeometric functions, Bessel functions, expo¬
nentials, and powers 828
7.68 Combinations of confluent hypergeometric functions and other special functions 832
7.69 Integration of confluent hypergeometric functions with respect to the index . . 834
7.7 Parabolic Cylinder Functions 835
7.71 Parabolic cylinder functions 835
7.72 Combinations of parabolic cylinder functions, powers, and exponentials 835
7.73 Combinations of parabolic cylinder and hyperbolic functions 837
7.74 Combinations of parabolic cylinder and trigonometric functions 837
7.75 Combinations of parabolic cylinder and Bessel functions 838
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric
functions 841
7.77 Integration of a parabolic cylinder function with respect to the index 842
7.8 Meijer s and MacRobert s Functions (G and E) 843
7.81 Combinations of the functions G and E and the elementary functions 843
7.82 Combinations of the functions G and E and Bessel functions 847
7.83 Combinations of the functions G and E and other special functions 840
8-9 Special Functions 851
8.1 Elliptic integrals and functions 851
8.11 Elliptic integrals 851
8.12 Functional relations between elliptic integrals 854
8.13 Elliptic functions 856
8.14 Jacobian elliptic functions 857
8.15 Properties of Jacobian elliptic functions and functional relationships between them 861
8.16 The Weierstrass function p(u) 865
8.17 The functions £(u) and cr(u) 868
8.18-8.19 Theta functions 869
8.2 The Exponential Integral Function and Functions Generated by It 875
8.21 The exponential integral function Ei(x) 875
8.22 The hyperbolic sine integral shix and the hyperbolic cosine integral cliix ... 878
8.23 The sine integral and the cosine integral: six and cix 878
8.24 The logarithm integral li(x) 879
8.25 The probability integral, the Fresnel integrals $(x), S(x), C{x), the error
function erf(x), and the complementary error function erfc(x) 879
8.26 Lobachevskiy s function L(x) 883
8.3 Euler s Integrals of the First and Second Kinds 883
8.31 The gamma function (Euler s integral of the second kind): T(z) 883
8.32 Representation of the gamma function as series and products 885
8.33 Functional relations involving the gamma function 886
8.34 The logarithm of the gamma function 888
8.35 The incomplete gamma function 890
8.36 The psi function ip(x) §92
8.37 The function (3(x) §96
8.38 The beta function (Euler s integral of the first kind): B(x,y) 897
8.39 The incomplete beta function q) 900
8.4-8.5 Bessel Functions and Functions Associated with Them 900
Definitions qqO
CONTENTS xiii
8.41 Integral representations of the functions Ju(z) and Nu(z) 901
8.42 Integral representations of the functions H^ z) and H^ z) 904
8.43 Integral representations of the functions Iu{z) and Ku(z) 900
8.44 Series representation 908
8.45 Asymptotic expansions of Bessel functions 909
8.46 Bessel functions of order equal to an integer plus one-half 913
8.47-8.48 Functional relations 915
8.49 Differential equations leading to Bessel functions 921
8.51-8.52 Series of Bessel functions 923
8.53 Expansion in products of Bessel functions 930
8.54 The zeros of Bessel functions 931
8.55 Struve functions 932
8.50 Thomson functions and their generalizations 934
8.57 Lommel functions 935
8.58 Anger and Weber functions Ju(z) and Eu(z) 938
8.59 Neumann s and Schlafli s polynomials: On(z) and Sn(z) 939
8.0 Mathieu Functions 940
8.00 Mathieu s equation 940
8.01 Periodic Mathieu functions 940
8.02 Recursion relations for the coefficients A^1^, -^2r+t^- ^4r+2^ • • • • 941
8.03 Mathieu functions with a purely imaginary argument 942
8.04 Non-periodic solutions of Mathieu s equation 943
8.05 Mathieu functions for negative q 943
8.06 Representation of Mathieu functions as series of Bessel functions 944
8.07 The general theory 947
8.7-8.8 Associated Legendre Functions 948
8.70 Introduction 948
8.71 Integral representations 950
8.72 Asymptotic series for large values of v 952
8.73-8.74 Functional relations 954
8.75 Special cases and particular values 957
8.76 Derivatives with respect to the order 959
8.77 Series representation 959
8.78 The zeros of associated Legendre functions 901
8.79 Series of associated Legendre functions 902
8.81 Associated Legendre functions with integral indices 904
8.82-8.83 Legendre functions 905
8.84 Conical functions 970
8.85 Toroidal functions 971
8.9 Orthogonal Polynomials 972
8.90 Introduction 972
8.91 Legendre polynomials 973
8.919 Series of products of Legendre and Chebyshev polynomials 977
8.92 Series of Legendre polynomials 978
8.93 Gegenbauer polynomials C*(t) 980
8.94 The Chebyshev polynomials Tn(x) and Un(x) 983
8.95 The Hermite polynomials Hn(x) 980
xiv
CONTENTS
8.96 Jacobi s polynomials -^8
8.97 The Laguerre polynomials
9.1 Hypergeometric Functions 995
9.10 Definition 995
9.11 Integral representations 995
9.12 Representation of elementary functions in terms of a hypergeometric functions . 995
9.13 Transformation formulas and the analytic continuation of functions defined by
hypergeometric series 998
9.14 A generalized hypergeometric series 1900
9.15 The hypergeometric differential equation 1000
9.16 Riemann s differential equation 1001
9.17 Representing the solutions to certain second-order differential equations using
a Riemann scheme 1007
9.18 Hypergeometric functions of two variables 1008
9.19 A hypergeometric function of several variables 1012
9.2 Confluent Hypergeometric Functions 1012
9.20 Introduction 1012
9.21 The functions 4 (a,7;z) and 4 (a,7;£) 1013
9.22-9.23 The Whittaker functions M t^(z) and tl(z) 1014
9.24-9.25 Parabolic cylinder functions Dp(z) 1018
9.26 Confluent hypergeometric series of two variables 1021
9.3 Meijer s G-Function 1022
9.30 Definition 1022
9.31 Functional relations 1023
9.32 A differential equation for the G-function 1024
9.33 Series of G-functions 1024
9.34 Connections with other special functions 1024
9.4 MacRobert s ^-Function 1025
9.41 Representation by means of multiple integrals 1025
9.42 Functional relations 1025
9.5 Riemann s Zeta Functions C(z,q), and £(z), and the Functions (z, .s, ?;) and f(s) 1026
9.51 Definition and integral representations 1026
9.52 Representation as a series or as an infinite product 1026
9.53 Functional relations 1027
9.54 Singular points and zeros 1028
9.55 The Lerch function 5, v) 1028
9.56 The function £ (s) 1029
9.6 Bernoulli numbers and polynomials, Euler numbers 1030
9.61 Bernoulli numbers 1030
9.62 Bernoulli polynomials 1031
9.63 Euler numbers 1032
9.64 The functions v(x), v{x,a), ti(x,f3), fi(x,(3,a), A{x,y) 1033
9.65 Euler polynomials 1033
9.7 Constants 1035
9.71 Bernoulli numbers 1035
9.72 Euler numbers 1035
9.73 Euler s and Catalan s constants 1036
5
1
CONTENTS xv
9.74 Stirling numbers 103G
10 Vector Field Theory 1039
10.1-10.8 Vectors, Vector Operators, and Integral Theorems 1039
10.11 Products of vectors 1039
10.12 Properties of scalar product 1039
10.13 Properties of vector product 1039
10.14 Differentiation of vectors 1039
10.21 Operators grad, div, and curl 1040
10.31 Properties of the operator V 1040
10.41 Solenoidal fields 1041
10.51 10.G1 Orthogonal curvilinear coordinates 1042
10.71 10.72 Vector integral theorems 1045
10.81 Integral rate of change theorems 1047
11 Algebraic Inequalities 1049
11.1 11.3 General Algebraic Inequalities 1049
11.11 Algebraic inequalities involving real numbers 1049
11.21 Algebraic inequalities involving complex numbers 1050
11.31 Inequalities for sets of complex numbers 1051
12 Integral Inequalities 1053
12.11 Mean value theorems 1053
12.111 First mean value theorem 1053
12.112 Second mean value theorem 1053
12.113 First mean value theorem for infinite integrals 1053
12.114 Second mean value theorem for infinite integrals 1054
12.21 Differentiation of definite integral containing a parameter 1054
12.211 Differentiation when limits are finite 1054
12.212 Differentiation when a limit is infinite 1054
12.31 Integral inequalities 1054
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals 1054
12.312 Holder s inequality for integrals 1054
12.313 Minkowski s inequality for integrals 1055
12.314 Chebyshev s inequality for integrals 1055
12.315 Young s inequality for integrals 1055
12.310 Steffensen s inequality for integrals 1055
12.317 Gram s inequality for integrals 1055
12.318 Ostrowski s inequality for integrals 1055
12.41 Convexity and Jensen s inequality 1050
12.411 Jensen s inequality. 105G
12.51 Fourier series and related inequalities 105G
12.511 Riemann-Lebesgue lemma 105G
12.512 Dirichlet lemma 1057
12.513 Parseval s theorem for trigonometric Fourier series 1057
12.514 Integral representation of the 7ith partial sum 1057
12.515 Generalized Fourier series 1057
12.51G Bessel s inequality for generalized Fourier series 1057
xvi
CONTENTS
12.517 Parseval s theorem for generalized Fourier series 1057
13 Matrices and related results 1059
13.11-13.12 Special matrices 1059
13.111 Diagonal matrix 1059
13.112 Identity matrix and null matrix 1059
13.113 Reducible and irreducible matrices 1059
13.114 Equivalent matrices 1059
13.115 Transpose of a matrix 1059
13.11G Adjoint matrix 1059
13.117 Inverse matrix 10G0
13.118 Trace of a matrix 10G0
13.119 Symmetric matrix 1000
13.120 Skew-symmetric matrix 1060
13.121 Triangular matrices 10G0
13.122 Orthogonal matrices 1000
13.123 Hermitian transpose of a matrix 10G0
13.124 Hermitian matrix 1000
13.125 Unitary matrix 1060
13.126 Eigenvalues and eigenvectors 1061
13.127 Nilpotent matrix 1061
13.128 Idempotent matrix 1061
13.129 Positive definite 1061
13.130 Non-negative definite 1061
13.131 Diagonally dominant 10G1
13.21 Quadratic forms 1061
13.211 Sylvester s law of inertia 1062
13.212 Rank 1062
13.213 Signature 10G2
13.214 Positive definite and semidefinite quadratic form 10G2
13.215 Basic theorems on quadratic forms 1062
13.31 Differentiation of matrices 1063
13.41 The matrix exponential 1004
3.411 Basic properties 10G4
14 Determinants 1065
14-11 Expansion of second- and third-order determinants 10G5
14.12 Basic properties 1Q05
14-13 Minors and cofactors of a determinant 1065
14*14 Principal minors 1000
14-15 Laplace expansion of a determinant
lilir , 1066
14.1b Jacobi s theorem lOgg
14-17 Hadamard s theorem 10G6
14*18 Hadamard s inequality -mG7
14-21 Cramer s rule ^ 1067
14-31 Some special determinants 1068
14.311 Vandermonde s determinant (alternant) lOgg
14.312 Circulants ^g
xvii
10G8
1069
10G9
10G9
1070
1071
1071
1071
1071
1071
1071
1071
1072
1072
1072
1072
1072
1072
1072
1072
1073
1073
1073
1074
1074
1074
1074
1075
107G
107G
107G
107G
107G
1077
1077
1077
1078
1078
1078
1078
1078
1078
1079
1079
1079
1079
Jacobian determinant
Hessian determinants
Wronskian determinants
Properties
Gram-Kowalewski theorem on linear dependence
Vector Norms
General properties
Principal vector norms
The norm ||x||i
The norm ||x||2 (Euclidean or £2 norm)
The norm ||x||oo
Matrix norms
General properties
Induced norms
Natural norm of unit matrix
Principal natural norms
Maximum absolute column sum norm
Spectral norm
Maximum absolute row sum norm
Spectral radius of a square matrix
Inequalities concerning matrix norms and the spectral radius
Deductions from Gerschgorin s theorem (see 15.814)
Inequalities involving eigenvalues of matrices
Cayley-Hamilton theorem
Corollaries
Inequalities for the characteristic polynomial
Named and unnamed inequalities
Parodi s theorem
Corollary of Brauer s theorem
Ballieu s theorem
Routh-Hurwitz theorem
Named theorems on eigenvalues
Schur s inequalities
Sturmian separation theorem
Poincare s separation theorem
Gerschgorin s theorem
Brauer s theorem
Perron s theorem
Frobenius theorem
Perron-Frobenius theorem
Wielandt s theorem
Ostrowski s theorem
First theorem due to Lyapunov
Second theorem due to Lyapunov
Hermitian matrices and diophantine relations involving circular functions of
rational angles due to Calogero and Perelomov
CONTENTS
xviii —
15.91 Variational principles
15.911 Rayleigh quotient
15.91*2 Basic theorems
16 Ordinary differential equations 1083
16.1-16.9 Results relating to the solution of ordinary differential equations 1083
16.11 First-order equations 1083
16.111 Solution of 3 first-order equation 1083
16.112 Cauchy problem 1083
16.113 Approximate solution to an equation 1083
16.114 Lipschitz continuity of a function 1084
16.21 Fundamental inequalities and related results 10S4
16.211 Gronwall s lemma 1081
16.212 Comparison of approximate solutions of a differential equation 1081
16.31 First-order systems 1085
16.311 Solution of a system of equations 108-)
16.312 Cauchy problem for a system 108 )
16.313 Approximate solution to a system 1085
16.314 Lipschitz continuity of a vector 1085
16.315 Comparison of approximate solutions of a system 1086
16.316 First-order linear differential equation 1086
16.317 Linear systems of differential equations 1086
16.41 Some special types of elementary differential equations 1087
16.411 Variables separable 1087
16.412 Exact differential equations 1087
16.413 Conditions for an exact equation 1087
16.414 Homogeneous differential equations 1087
16.51 Second-order equations 1088
16.511 Adjoint and self-adjoint equations 1088
16.512 Abel s identity 1088
16.513 Lagrange identity 1089
16.514 The Riccati equation 1089
16.515 Solutions of the Riccati equation 1089
16.516 Solution of a second-order linear differential equation 1090
16.61-16.62 Oscillation and non-oscillation theorems for second-order equations 1090
16.611 First basic comparison theorem 1090
16.622 Second basic comparison theorem 1091
16.623 Interlacing of zeros 1091
16.624 Sturm separation theorem 1091
16.625 Sturm comparison theorem 1091
16.626 Szegd s comparison theorem 1091
16.627 Picone s identity 1092
16.628 Sturm-Picone theorem 1092
16.629 Oscillation on the half line 1092
16.71 Two related comparison theorems 1093
16.711 Theorem 1 1093
16.712 Theorem 2 1093
1G.81-16.82 Non-oscillatory solutions 1093
X
CONTENTS xix
10.811 Kneser s non-oscillation theorem 1094
10.822 Comparison theorem for non-oscillation 1091
10.823 Necessary and sufficient conditions for non-oscillation 3094
10.91 Some growth estimates for solutions of second-order equations 1094
10.911 Strictly increasing and decreasing solutions 1094
1G.912 General result on dominant and subdominant solutions 1095
10.913 Estimate of dominant solution 1095
10.914 A theorem due to Lyapunov 1090
10.92 Boundedness theorems 109G
10.921 All solutions of the equation 1090
1G.922 If all solutions of the equation 109G
10.923 If a(x) —? oc monotonically as x —» cxd, then all solutions of 1090
1G.924 Consider the equation 1090
10.93 Growth of maxima of y 1097
17 Fourier, Laplace, and Mellin Transforms 1099
17.1 17.4 Integral Transforms 1099
17.11 Laplace transform 1099
17.12 Basic properties of the Laplace transform 1099
17.13 Table of Laplace transform pairs 1100
17.21 Fourier transform 1109
17.22 Basic properties of the Fourier transform 1110
17.23 Table of Fourier transform pairs 1110
17.24 Table of Fourier transform pairs for spherically symmetric functions 1112
17.31 Fourier sine and cosine transforms 1113
17.32 Basic properties of the Fourier sine and cosine transforms 1113
17.33 Table of Fourier sine transforms 1114
17.34 Table of Fourier cosine transforms 1118
17.35 Relationships between transforms 1121
17.41 Mellin transform H21
17.42 Basic properties of the Mellin transform 1122
17.43 Table of Mellin cosine transforms 1122
18 The z-transform 1127
18.1-18.3 Definition, Bilateral, and Unilateral --Transforms 1127
18.1 Definitions H27
18.2 Bilateral ^-transform 1127
18.3 Unilateral z-transform 1129
References ** ***
Supplemental references 11
Function and constant index H l l
General index ^ ^
|
| any_adam_object | 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 | BV013376668 |
| callnumber-first | Q - Science |
| callnumber-label | QA55 |
| callnumber-raw | QA55 |
| callnumber-search | QA55 |
| callnumber-sort | QA 255 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SH 500 |
| classification_tum | MAT 001k MAT 260k |
| ctrlnum | (OCoLC)247849178 (DE-599)BVBBV013376668 |
| 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 3212 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 6. ed. |
| format | Book |
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| genre_facet | Formelsammlung Tabelle Tabelle - Spezielle Funktion |
| id | DE-604.BV013376668 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:44:44Z |
| institution | BVB |
| isbn | 0122947576 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009123705 |
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| physical | XLVII, 1163 S. |
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| publisher | Acad. 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 6. ed. San Diego [u.a.] Acad. Press 2000 XLVII, 1163 S. txt rdacontent n rdamedia nc rdacarrier Auch als CD-ROM-Ausg. u.d.T.: Table of integrals, series, and products. - Aus dem Russ. übers. Produkt Mathematik (DE-588)4126359-5 gnd rswk-swf Integral (DE-588)4131477-3 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Produkt (DE-588)4139399-5 gnd rswk-swf Reihe Musik (DE-588)4368335-6 gnd rswk-swf Reihe (DE-588)4049197-3 gnd rswk-swf Summe (DE-588)4193845-8 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content (DE-588)4184303-4 Tabelle gnd-content Tabelle - Spezielle Funktion 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 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009123705&sequence=000001&line_number=0001&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 Produkt Mathematik (DE-588)4126359-5 gnd Integral (DE-588)4131477-3 gnd Mathematik (DE-588)4037944-9 gnd Produkt (DE-588)4139399-5 gnd Reihe Musik (DE-588)4368335-6 gnd Reihe (DE-588)4049197-3 gnd Summe (DE-588)4193845-8 gnd |
| subject_GND | (DE-588)4126359-5 (DE-588)4131477-3 (DE-588)4037944-9 (DE-588)4139399-5 (DE-588)4368335-6 (DE-588)4049197-3 (DE-588)4193845-8 (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_full | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik |
| title_fullStr | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik |
| title_full_unstemmed | Table of integrals, series, and products I. S. Gradshteyn and I. M. Ryzhik |
| title_short | Table of integrals, series, and products |
| title_sort | table of integrals series and products |
| topic | Produkt Mathematik (DE-588)4126359-5 gnd Integral (DE-588)4131477-3 gnd Mathematik (DE-588)4037944-9 gnd Produkt (DE-588)4139399-5 gnd Reihe Musik (DE-588)4368335-6 gnd Reihe (DE-588)4049197-3 gnd Summe (DE-588)4193845-8 gnd |
| topic_facet | Produkt Mathematik Integral Mathematik Produkt Reihe Musik Reihe Summe Formelsammlung Tabelle Tabelle - Spezielle Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009123705&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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