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
02 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
03 Functional series 15
0.30 Definitions and theorems 15
0.31 Power series 16
0.32 Fourier series 18
0.33 Asymptotic series 20
04 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 26
1.2 The Exponential Function 26
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”П（x） 49
1.49 The hyperbolic amplitude（the Gudermannian）gd x 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
1.62-163 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 α+βx 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 α+βx 79
2.2 Algebraic functions 80
2.20 Introduction 80
2.21 Forms containing the binomial a+bxk and？ 81
2.22-2.23 Forms containing？ 83
2.24 Forms containing？and the binomial o+βx 86
2.25 Forms containing？ 90
2.26 Forms containing？and integral powers of x 92
2.27 Forms containing？and integral powers of x 97
2.28 Forms containing？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 sinh x,cosh x,tanh x,and coth x 105
2.44-2.45 Rational functions of hyperbolic functions 121
2.46 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
2.5-2.6 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？ 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
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（α+βx） 312
3.24-3.27 Powers of x,of binomials of the form α+βxp 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
3.36-3.37 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？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
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 x 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 601
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 603
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
4.63-464 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 622
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 623
5.5 Bessel Functions 624
6-7 Definite Integrals of Special Functions 625
6.1 Elliptic Integrals and Functions 625
6.11 Forms containing F（x,k） 625
6.12 Forms containing E（x,k） 626
6.13 Integration of elliptic integrals with respect to the modulus 626
6.14-6.15 Complete elliptic integrals 626
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 functionψ（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
6.84-6.85 Combinations of Struve and Bessel functions 748
6.86 Lommel functions 752
6.87 Thomson functions 754
6.9 Mathieu Functions 755
6.91 Mathieu functions 755
6.92 Combinations of Mathieu,hyperbolic,and trigonometric functions 756
6.93 Combinations of Mathieu and Bessel functions 759
6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems 759
7.1-7.2 Associated Legendre Functions 762
7.11 Associated Legendre functions 762
7.12-7.13 Combinations of associated Legendre functions and powers 763
7.14 Combinations of associated Legendre functions,exponentials,and powers 769
7.15 Combinations of associated Legendre and hyperbolic functions 771
7.16 Combinations of associated Legendre functions,powers,and trigonometric functions 772
7.17 A combination of an associated Legendre function and the probability integral 774
7.18 Combinations of associated Legendre and Bessel functions 774
7.19 Combinations of associated Legendre functions and functions generated by Bessel functions 780
7.21 Integration of associated Legendre functions with respect to the order 781
7.22 Combinations of Legendre polynomials,rational functions,and algebraic functions 782
7.23 Combinations of Legendre polynomials and powers 784
7.24 Combinations of Legendre polynomials and other elementary functions 785
7.25 Combinations of Legendre polynomials and Bessel functions 787
7.3-7.4 Orthogonal Polynomials 788
7.31 Combinations of Gegenbauer polynomials Cv n（x）and powers 788
7.32 Combinations of Gegenbauer polynomials Cv n（x）and some elementary functions 790
7.33 Combinations of the polynomials Cv n（x）and Bessel functions.Integration of Gegenbauer functions with respect to the index 791
7.34 Combinations of Chebyshev polynomials and powers 793
7.35 Combinations of Chebyshev polynomials and some elementary functions 794
7.36 Combinations of Chebyshev polynomials and Bessel functions 795
7.37-7.38 Hermite polynomials 796
7.39 Jacobi polynomials 800
7.41-7.42 Laguerre polynomials 801
7.5 Hypergeometric Functions 806
7.51 Combinations of hypergeometric functions and powers 806
7.52 Combinations of hypergeometric functions and exponentials 807
7.53 Hypergeometric and trigonometric functions 810
7.54 Combinations of hypergeometric and Bessel functions 810
7.6 Confluent Hypergeometric Functions 814
7.61 Combinations of confluent hypergeometric functions and powers 814
7.62-7.63 Combinations of confluent hypergeometric functions and exponentials 815
7.64 Combinations of confluent hypergeometric and trigonometric functions 822
7.65 Combinations of confluent hypergeometric functions and Bessel functions 824
7.66 Combinations of confluent hypergeometric functions,Bessel functions,and powers 824
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 849
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 ？（u） 865
8.17 The functions ζ（u） and σ（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 shi x and the hyperbolic cosine integral chi x 878
8.23 The sine integral and the cosine integral：six and ci x 878
8.24 The logarithm integral li（x） 879
8.25 The probability integral,the Fresnel integralsφ（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）：Γ（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 functionψ（x） 892
8.37 The functionβ（x） 896
8.38 The beta function（Euler's integral of the first kind）：B（x,y） 897
8.39 The incomplete beta function Bx（p,q） 900
8.4-8.5 Bessel Functions and Functions Associated with Them 900
8.40 Definitions 900
8.41 Integral representations of the functions Jv（z）and Nv（z） 901
8.42 Integral representations of the functions H（1）v（z）and H（2）v（z） 904
8.43 Integral representations of the functions Iv（z）and Kv（z） 906
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
8.56 Thomson functions and their generalizations 934
8.57 Lommel functions 935
8.58 Anger and Weber functions Jv（z）and Ev（z） 938
8.59 Neumann's and Schl？fli's polynomials：On（z）and Sn（z） 939
8.6 Mathieu Functions 940
8.60 Mathieu's equation 940
8.61 Periodic Mathieu functions 940
8.62 Recursion relations for the coefficients A2（2n）2r,A（2n+1）2r+1,B（2n+1）2r+1,B（2n+2）2r+2 941
8.63 Mathieu functions with a purely imaginary argument 942
8.64 Non-periodic solutions of Mathieu's equation 943
8.65 Mathieu functions for negative q 943
8.66 Representation of Mathieu functions as series of Bessel functions 944
8.67 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 961
8.79 Series of associated Legendre functions 962
8.81 Associated Legendre functions with integral indices 964
8.82-8.83 Legendre functions 965
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 Cλn（t） 980
8.94 The Chebyshev polynomials Tn（x）and Un（x） 983
8.95 The Hermite polynomials Hn（x） 986
8.96 Jacobi's polynomials 988
8.97 The Laguerre polynomials 990
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 1000
9.15 The hypergeometric differential equation 1000
9.16 Riemann's differential equation 1004
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 functionsφ（α，γ；z）andψ（α，γ；z） 1013
9.22-9.23 The Whittaker functions Mλ，μ（z）and Wλ，μ（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 E-Function 1025
9.41 Representation by means of multiple integrals 1025
9.42 Functional relations 1025
9.5 Riemann's Zeta Functionsζ（z.q）andζ（z）and the Functionsφ（z,s,v）and ξ（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 functionφ（z,s,v） 1028
9.56 The functionξ（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.α），μ（x，β），μ（x，β，α）．λ（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
9.74 Stirling numbers 1036
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 ？ 1040
10.41 Solenoidal fields 1041
10.51-10.61 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 Cauch-Schwarz-Buniakowsky inequality for integrals 1054
12.312 H？lder'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
12.316 Steffensen's inequality for integrals 1055
12.317 Grams inequality for integrals 1055
12.318 Ostrowski's inequality for integrals 1055
12.41 Convexity and Jensen's inequality 1056
12.411 Jensen's inequality 1056
12.51 Fourier series and related inequalities 1056
12.511 Riemann-Lebesgue lemma 1056
12.512 Dirichlet lemma 1057
12.513 Parseval's theorem for trigonometric Fourier series 1057
12.514 Integral representation of the nth partial sum 1057
12.515 Generalized Fourier series 1057
12.516 Bessel's inequality for generalized Fourier series 1057
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
13.116 Adjoint matrix 1059
13.117 Inverse matrix 1060
13.118 Trace of a matrix 1060
13.119 Symmetric matrix 1060
13.120 Skew-symmetric matrix 1060
13.121 Triangular matrices 1060
13.122 Orthogonal matrices 1060
13.123 Hermitian transpose of a matrix 1060
13.124 Hermitian matrix 1060
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 1061
13.21 Quadratic forms 1061
13.211 Sylvester's law of inertia 1062
13.212 Rank 1062
13.213 Signature 1062
13.214 Positive definite and semidefinite quadratic form 1062
13.215 Basic theorems on quadratic forms 1062
13.31 Differentiation of matrices 1063
13.41 The matrix exponential 1064
3.411 Basic properties 1064
14 Determinants 1065
14.11 Expansion of second- and third-order determinants 1065
14.12 Basic properties 1065
14.13 Minors and cofactors of a determinant 1065
14.14 Principal minors 1066
14.15 Laplace expansion of a determinant 1066
14.16 Jacobi's theorem 1066
14.17 Hadamard's theorem 1066
14.18 Hadamard's inequality 1067
14.21 Cramer's rule 1067
14.31 Some special determinants 1068
14.311 Vandermonde's determinant（alternant） 1068
14.312 Circulants 1068
14.313 Jacobian determinant 1068
14.314 Hessian determinants 1069
14.315 Wronskian determinants 1069
14.316 Properties 1069
14.317 Gram-Kowalewski theorem on linear dependence 1070
15 Norms 1071
15.1-15.9 Vector Norms 1071
15.11 General properties 1071
15.21 Principal vector norms 1071
15.211 The norm ‖x‖1 1071
15.212 The norm ‖x‖2（Euclidean or L2 norm） 1071
15.213 The norm ‖x‖∞ 1071
15.31 Matrix norms 1072
15.311 General properties 1072
15.312 Induced norms 1072
15.313 Natural norm of unit matrix 1072
15.41 Principal natural norms 1072
15.411 Maximum absolute column sum norm 1072
15.412 Spectral norm 1072
15.413 Maximum absolute row sum norm 1072
15.51 Spectral radius of a square matrix 1073
15.511 Inequalities concerning matrix norms and the spectral radius 1073
15.512 Deductions from Gerschgorin's theorem（see 15.814） 1073
15.61 Inequalities involving eigenvalues of matrices 1074
15.611 Cayley-Hamilton theorem 1074
15.612 Corollaries 1074
15.71 Inequalities for the characteristic polynomial 1074
15.711 Named and unnamed inequalities 1075
15.712 Parodi's theorem 1076
15.713 Corollary of Brauer's theorem 1076
15.714 Ballieus theorem 1076
15.715 Routh-Hurwitz theorem 1076
15.81-15.82 Named theorems on eigenvalues 1076
15.811 Schur's inequalities 1077
15.812 Sturmian separation theorem 1077
15.813 Poincare's separation theorem 1077
15.814 Gerschgorin's theorem 1078
15.815 Brauers theorem 1078
15.816 Perron's theorem 1078
15.817 Frobenius theorem 1078
15.818 Perron-Frobenius theorem 1078
15.819 Wielandt's theorem 1078
15.820 Ostrowskis theorem 1079
15.821 First theorem due to Lyapunov 1079
15.822 Second theorem due to Lyapunov 1079
15.823 Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov 1079
15.91 Variational principles 1081
15.911 Rayleigh quotient 1081
15.912 Basic theorems 1081
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 a 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 1084
16.211 Gronwall's lemma 1084
16.212 Comparison of approximate solutions of a differential equation 1084
16.31 Firstorder systems 1085
16.311 Solution of a system of equations 1085
16.312 Cauchy problem for a system 1085
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-orderlinear differential equation 1090
16.61-16.62 Oscillation and non-oscillation theorems for secondorder 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 Szeg？'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 Theorem1 1093
16.712 Theorem2 1093
16.81-16.82 Non-oscillatory solutions 1093
16.811 Kneser's non-oscillation theorem 1094
16.822 Comparison theorem for non-oscillation 1094
16.823 Necessary and sufficient conditions for non-oscillation 1094
16.91 Some growth estimates for solutions of second-order equations 1094
16.911 Strictly increasing and decreasing solutions 1094
16.912 General result on dominant and subdominant solutions 1095
16.913 Estimate of dominant solution 1095
16.914 A theorem due to Lyapunov 1096
16.92 Boundedness theorems 1096
16.921 All solutions of the equation 1096
16.922 If all solutions of the equation 1096
16.923 If a（x）→∞monotonically as x→∞，then all solutions of 1096
16.924 Consider the equation 1096
16.93 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 1121
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 z-Transforms 1127
18.1 Definitions 1127
18.2 Bilateral z-transform 1127
18.3 Unilateral z-transform 1129
References 1133
Supplemental references 1137
Function and constant index 1143
General index 1153