积分、级数和乘积表 第6版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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0 Introduction1

0.1 Finite sums1

0.11 Progressions1

0.12 Sums of powers of natural numbers1

0.13 Sums of reciprocals of natural numbers2

0.14 Sums of products of reciprocals of natural numbers3

0.15 Sums of the binomial coefficients3

02 Numerical series and infinite products6

0.21 The convergence of numerical series6

0.22 Convergence tests6

0.23-0.24 Examples of numerical series8

0.25 Infinite products14

0.26 Examples of infinite products14

03 Functional series15

0.30 Definitions and theorems15

0.31 Power series16

0.32 Fourier series18

0.33 Asymptotic series20

04 Certain formulas from differential calculus21

0.41 Differentiation of a definite integral with respect to a parameter21

0.42 The nth derivative of a product（Leibniz's rule）21

0.43 The nth derivative of a composite function21

0.44 Integration by substitution23

1 Elementary Functions25

1.1 Power of Binomials25

1.11 Power series25

1.12 Series of rational fractions26

1.2 The Exponential Function26

1.21 Series representation26

1.22 Functional relations27

1.23 Series of exponentials27

1.3-1.4 Trigonometric and Hyperbolic Functions27

1.30 Introduction28

1.31 The basic functional relations28

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 functions32

1.34 Certain sums of trigonometric and hyperbolic functions35

1.35 Sums of powers of trigonometric functions of multiple angles36

1.36 Sums of products of trigonometric functions of multiple angles37

1.37 Sums of tangents of multiple angles38

1.38 Sums leading to hyperbolic tangents and cotangents38

1.39 The representation of cosines and sines of multiples of the angle as finite products39

1.41 The expansion of trigonometric and hyperbolic functions in power series41

1.42 Expansion in series of simple fractions42

1.43 Representation in the form of an infinite product43

1.44-1.45 Trigonometric（Fourier）series44

1.46 Series of products of exponential and trigonometric functions48

1.47 Series of hyperbolic functions49

1.48 Lobachevskiy's“Angle of parallelism”П（x）49

1.49 The hyperbolic amplitude（the Gudermannian）gd x50

1.5 The Logarithm51

1.51 Series representation51

1.52 Series of logarithms（cf.1.431）53

1.6 The Inverse Trigonometric and Hyperbolic Functions54

1.61 The domain of definition54

1.62-163 Functional relations54

1.64 Series representations58

2 Indefinite Integrals of Elementary Functions61

2.0 Introduction61

2.00 General remarks61

2.01 The basic integrals61

2.02 General formulas62

2.1 Rational functions64

2.10 General integration rules64

2.11-2.13 Forms containing the binomial a+bxk66

2.14 Forms containing the binomial 1±xn72

2.15 Forms containing pairs of binomials：a+bx and α+βx76

2.16 Forms containing the trinomial a+bxk+cx2k76

2.17 Forms containing the quadratic trinomial a+bx+cx2 and powers of x77

2.18 Forms containing the quadratic trinomial a+bx+cx2 and the binomial α+βx79

2.2 Algebraic functions80

2.20 Introduction80

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+βx86

2.25 Forms containing？90

2.26 Forms containing？and integral powers of x92

2.27 Forms containing？and integral powers of x97

2.28 Forms containing？and first-and second-degree polynomials101

2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals102

2.3 The Exponential Function104

2.31 Forms containing eax104

2.32 The exponential combined with rational functions of x104

2.4 Hyperbolic Functions105

2.41-2.43 Powers of sinh x,cosh x,tanh x,and coth x105

2.44-2.45 Rational functions of hyperbolic functions121

2.46 Algebraic functions of hyperbolic functions128

2.47 Combinations of hyperbolic functions and powers136

2.48 Combinations of hyperbolic functions,exponentials,and powers145

2.5-2.6 Trigonometric Functions147

2.50 Introduction147

2.51-2.52 Powers of trigonometric functions147

2.53-2.54 Sines and cosines of multiple angles and of linear and more complicated func-tions of the argument157

2.55-2.56 Rational functions of the sine and cosine167

2.57 Integrals containing？175

2.58-2.62 Integrals reducible to elliptic and pseudo-elliptic integrals180

2.63-2.65 Products of trigonometric functions and powers210

2.66 Combinations of trigonometric functions and exponentials222

2.67 Combinations of trigonometric and hyperbolic functions227

2.7 Logarithms and Inverse-Hyperbolic Functions233

2.71 The logarithm233

2.72-2.73 Combinations of logarithms and algebraic functions233

2.74 Inverse hyperbolic functions236

2.8 Inverse Trigonometric Functions237

2.81 Arcsines and arccosines237

2.82 The arcsecant,the arccosecant,the arctangent and the arccotangent238

2.83 Combinations of arcsine or arccosine and algebraic functions238

2.84 Combinations of the arcsecant and arccosecant with powers of x240

2.85 Combinations of the arctangent and arccotangent with algebraic functions240

3-4 Definite Integrals of Elementary Functions243

3.0 Introduction243

3.01 Theorems of a general nature243

3.02 Change of variable in a definite integral244

3.03 General formulas245

3.04 Improper integrals247

3.05 The principal values of improper integrals248

3.1-3.2 Power and Algebraic Functions248

3.11 Rational functions249

3.12 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials249

3.13-3.17 Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with rational functions250

3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions310

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 x319

3.3-3.4 Exponential Functions331

3.31 Exponential functions331

3.32-3.34 Exponentials of more complicated arguments333

3.35 Combinations of exponentials and rational functions336

3.36-3.37 Combinations of exponentials and algebraic functions340

3.38-3.39 Combinations of exponentials and arbitrary powers342

3.41-3.44 Combinations of rational functions of powers and exponentials349

3.45 Combinations of powers and algebraic functions of exponentials358

3.46-3.48 Combinations of exponentials of more complicated arguments and powers360

3.5 Hyperbolic Functions365

3.51 Hyperbolic functions366

3.52-3.53 Combinations of hyperbolic functions and algebraic functions369

3.54 Combinations of hyperbolic functions and exponentials376

3.55-3.56 Combinations of hyperbolic functions,exponentials,and powers380

3.6-4.1 Trigonometric Functions384

3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles385

3.62 Powers of trigonometric functions388

3.63 Powers of trigonometric functions and trigonometric functions of linear functions390

3.64-3.65 Powers and rational functions of trigonometric functions395

3.66 Forms containing powers of linear functions of trigonometric functions399

3.67 Square roots of expressions containing trigonometric functions402

3.68 Various forms of powers of trigonometric functions404

3.69-3.71 Trigonometric functions of more complicated arguments408

3.72-3.74 Combinations of trigonometric and rational functions417

3.75 Combinations of trigonometric and algebraic functions428

3.76-3.77 Combinations of trigonometric functions and powers429

3.78-3.81 Rational functions of x and of trigonometric functions440

3.82-3.83 Powers of trigonometric functions combined with other powers453

3.84 Integrals containing？and similar expressions466

3.85-3.88 Trigonometric functions of more complicated arguments combined with powers469

3.89-3.91 Trigonometric functions and exponentials479

3.92 Trigonometric functions of more complicated arguments combined with expo-nentials487

3.93 Trigonometric and exponential functions of trigonometric functions490

3.94-3.97 Combinations involving trigonometric functions,exponentials,and powers492

3.98-3.99 Combinations of trigonometric and hyperbolic functions504

4.11-4.12 Combinations involving trigonometric and hyperbolic functions and powers511

4.13 Combinations of trigonometric and hyperbolic functions and exponentials517

4.14 Combinations of trigonometric and hyperbolic functions,exponentials,and powers520

4.2-4.4 Logarithmic Functions522

4.21 Logarithmic functions522

4.22 Logarithms of more complicated arguments525

4.23 Combinations of logarithms and rational functions530

4.24 Combinations of logarithms and algebraic functions532

4.25 Combinations of logarithms and powers534

4.26-4.27 Combinations involving powers of the logarithm and other powers537

4.28 Combinations of rational functions of In x and powers549

4.29-4.32 Combinations of logarithmic functions of more complicated arguments and powers551

4.33-4.34 Combinations of logarithms and exponentials567

4.35-4.36 Combinations of logarithms,exponentials,and powers568

4.37 Combinations of logarithms and hyperbolic functions574

4.38-4.41 Logarithms and trigonometric functions577

4.42-4.43 Combinations of logarithms,trigonometric functions,and powers590

4.44 Combinations of logarithms,trigonometric functions,and exponentials595

4.5 Inverse Trigonometric Functions596

4.51 Inverse trigonometric functions596

4.52 Combinations of arcsines,arccosines,and powers596

4.53-4.54 Combinations of arctangents,arccotangents,and powers597

4.55 Combinations of inverse trigonometric functions and exponentials601

4.56 A combination of the arctangent and a hyperbolic function601

4.57 Combinations of inverse and direct trigonometric functions601

4.58 A combination involving an inverse and a direct trigonometric function and a power603

4.59 Combinations of inverse trigonometric functions and logarithms603

4.6 Multiple Integrals604

4.60 Change of variables in multiple integrals604

4.61 Change of the order of integration and change of variables604

4.62 Double and triple integrals with constant limits607

4.63-464 Multiple integrals609

5 Indefinite Integrals of Special Functions615

5.1 Elliptic Integrals and Functions615

5.11 Complete elliptic integrals615

5.12 Elliptic integrals616

5.13 Jacobian elliptic functions618

5.14 Weierstrass elliptic functions622

5.2 The Exponential Integral Function622

5.21 The exponential integral function622

5.22 Combinations of the exponential integral function and powers622

5.23 Combinations of the exponential integral and the exponential622

5.3 The Sine Integral and the Cosine Integral623

5.4 The Probability Integral and Fresnel Integrals623

5.5 Bessel Functions624

6-7 Definite Integrals of Special Functions625

6.1 Elliptic Integrals and Functions625

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 modulus626

6.14-6.15 Complete elliptic integrals626

6.16 The theta function627

6.17 Generalized elliptic integrals628

6.2-6.3 The Exponential Integral Function and Functions Generated by It630

6.21 The logarithm integral630

6.22-6.23 The exponential integral function631

6.24-6.26 The sine integral and cosine integral functions633

6.27 The hyperbolic sine integral and hyperbolic cosine integral functions638

6.28-6.31 The probability integral638

6.32 Fresnel integrals642

6.4 The Gamma Function and Functions Generated by It644

6.41 The gamma function644

6.42 Combinations of the gamma function,the exponential,and powers645

6.43 Combinations of the gamma function and trigonometric functions648

6.44 The logarithm of the gamma function649

6.45 The incomplete gamma function650

6.46-6.47 The functionψ（x）651

6.5-6.7 Bessel Functions652

6.51 Bessel functions653

6.52 Bessel functions combined with x and x2657

6.53-6.54 Combinations of Bessel functions and rational functions662

6.55 Combinations of Bessel functions and algebraic functions666

6.56-6.58 Combinations of Bessel functions and powers667

6.59 Combinations of powers and Bessel functions of more complicated arguments681

6.61 Combinations of Bessel functions and exponentials686

6.62-6.63 Combinations of Bessel functions,exponentials,and powers691

6.64 Combinations of Bessel functions of more complicated arguments,exponentials,and powers701

6.65 Combinations of Bessel and exponential functions of more complicated argu-ments and powers703

6.66 Combinations of Bessel,hyperbolic,and exponential functions705

6.67-6.68 Combinations of Bessel and trigonometric functions709

6.69-6.74 Combinations of Bessel and trigonometric functions and powers719

6.75 Combinations of Bessel,trigonometric,and exponential functions and powers735

6.76 Combinations of Bessel,trigonometric,and hyperbolic functions739

6.77 Combinations of Bessel functions and the logarithm,or arctangent739

6.78 Combinations of Bessel and other special functions740

6.79 Integration of Bessel functions with respect to the order741

6.8 Functions Generated by Bessel Functions745

6.81 Struve functions745

6.82 Combinations of Struve functions,exponentials,and powers747

6.83 Combinations of Struve and trigonometric functions748

6.84-6.85 Combinations of Struve and Bessel functions748

6.86 Lommel functions752

6.87 Thomson functions754

6.9 Mathieu Functions755

6.91 Mathieu functions755

6.92 Combinations of Mathieu,hyperbolic,and trigonometric functions756

6.93 Combinations of Mathieu and Bessel functions759

6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems759

7.1-7.2 Associated Legendre Functions762

7.11 Associated Legendre functions762

7.12-7.13 Combinations of associated Legendre functions and powers763

7.14 Combinations of associated Legendre functions,exponentials,and powers769

7.15 Combinations of associated Legendre and hyperbolic functions771

7.16 Combinations of associated Legendre functions,powers,and trigonometric functions772

7.17 A combination of an associated Legendre function and the probability integral774

7.18 Combinations of associated Legendre and Bessel functions774

7.19 Combinations of associated Legendre functions and functions generated by Bessel functions780

7.21 Integration of associated Legendre functions with respect to the order781

7.22 Combinations of Legendre polynomials,rational functions,and algebraic functions782

7.23 Combinations of Legendre polynomials and powers784

7.24 Combinations of Legendre polynomials and other elementary functions785

7.25 Combinations of Legendre polynomials and Bessel functions787

7.3-7.4 Orthogonal Polynomials788

7.31 Combinations of Gegenbauer polynomials Cv n（x）and powers788

7.32 Combinations of Gegenbauer polynomials Cv n（x）and some elementary functions790

7.33 Combinations of the polynomials Cv n（x）and Bessel functions.Integration of Gegenbauer functions with respect to the index791

7.34 Combinations of Chebyshev polynomials and powers793

7.35 Combinations of Chebyshev polynomials and some elementary functions794

7.36 Combinations of Chebyshev polynomials and Bessel functions795

7.37-7.38 Hermite polynomials796

7.39 Jacobi polynomials800

7.41-7.42 Laguerre polynomials801

7.5 Hypergeometric Functions806

7.51 Combinations of hypergeometric functions and powers806

7.52 Combinations of hypergeometric functions and exponentials807

7.53 Hypergeometric and trigonometric functions810

7.54 Combinations of hypergeometric and Bessel functions810

7.6 Confluent Hypergeometric Functions814

7.61 Combinations of confluent hypergeometric functions and powers814

7.62-7.63 Combinations of confluent hypergeometric functions and exponentials815

7.64 Combinations of confluent hypergeometric and trigonometric functions822

7.65 Combinations of confluent hypergeometric functions and Bessel functions824

7.66 Combinations of confluent hypergeometric functions,Bessel functions,and powers824

7.67 Combinations of confluent hypergeometric functions,Bessel functions,expo-nentials,and powers828

7.68 Combinations of confluent hypergeometric functions and other special functions832

7.69 Integration of confluent hypergeometric functions with respect to the index834

7.7 Parabolic Cylinder Functions835

7.71 Parabolic cylinder functions835

7.72 Combinations of parabolic cylinder functions,powers,and exponentials835

7.73 Combinations of parabolic cylinder and hyperbolic functions837

7.74 Combinations of parabolic cylinder and trigonometric functions837

7.75 Combinations of parabolic cylinder and Bessel functions838

7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions841

7.77 Integration of a parabolic cylinder function with respect to the index842

7.8 Meijer's and MacRobert's Functions（G and E）843

7.81 Combinations of the functions G and E and the elementary functions843

7.82 Combinations of the functions G and E and Bessel functions847

7.83 Combinations of the functions G and E and other special functions849

8-9 Special Functions851

8.1 Elliptic integrals and functions851

8.11 Elliptic integrals851

8.12 Functional relations between elliptic integrals854

8.13 Elliptic functions856

8.14 Jacobian elliptic functions857

8.15 Properties of Jacobian elliptic functions and functional relationships between them861

8.16 The Weierstrass function ？（u）865

8.17 The functions ζ（u） and σ（u）868

8.18-8.19 Theta functions869

8.2 The Exponential Integral Function and Functions Generated by It875

8.21 The exponential integral function Ei（x）875

8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x878

8.23 The sine integral and the cosine integral：six and ci x878

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 Kinds883

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 products885

8.33 Functional relations involving the gamma function886

8.34 The logarithm of the gamma function888

8.35 The incomplete gamma function890

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 Them900

8.40 Definitions900

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 representation908

8.45 Asymptotic expansions of Bessel functions909

8.46 Bessel functions of order equal to an integer plus one-half913

8.47-8.48 Functional relations915

8.49 Differential equations leading to Bessel functions921

8.51-8.52 Series of Bessel functions923

8.53 Expansion in products of Bessel functions930

8.54 The zeros of Bessel functions931

8.55 Struve functions932

8.56 Thomson functions and their generalizations934

8.57 Lommel functions935

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 Functions940

8.60 Mathieu's equation940

8.61 Periodic Mathieu functions940

8.62 Recursion relations for the coefficients A2（2n）2r,A（2n+1）2r+1,B（2n+1）2r+1,B（2n+2）2r+2941

8.63 Mathieu functions with a purely imaginary argument942

8.64 Non-periodic solutions of Mathieu's equation943

8.65 Mathieu functions for negative q943

8.66 Representation of Mathieu functions as series of Bessel functions944

8.67 The general theory947

8.7-8.8 Associated Legendre Functions948

8.70 Introduction948

8.71 Integral representations950

8.72 Asymptotic series for large values of｜v｜952

8.73-8.74 Functional relations954

8.75 Special cases and particular values957

8.76 Derivatives with respect to the order959

8.77 Series representation959

8.78 The zeros of associated Legendre functions961

8.79 Series of associated Legendre functions962

8.81 Associated Legendre functions with integral indices964

8.82-8.83 Legendre functions965

8.84 Conical functions970

8.85 Toroidal functions971

8.9 Orthogonal Polynomials972

8.90 Introduction972

8.91 Legendre polynomials973

8.919 Series of products of Legendre and Chebyshev polynomials977

8.92 Series of Legendre polynomials978

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 polynomials988

8.97 The Laguerre polynomials990

9.1 Hypergeometric Functions995

9.10 Definition995

9.11 Integral representations995

9.12 Representation of elementary functions in terms of a hypergeometric functions995

9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series998

9.14 A generalized hypergeometric series1000

9.15 The hypergeometric differential equation1000

9.16 Riemann's differential equation1004

9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme1007

9.18 Hypergeometric functions of two variables1008

9.19 A hypergeometric function of several variables1012

9.2 Confluent Hypergeometric Functions1012

9.20 Introduction1012

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 variables1021

9.3 Meijer's G-Function1022

9.30 Definition1022

9.31 Functional relations1023

9.32 A differential equation for the G-function1024

9.33 Series of G-functions1024

9.34 Connections with other special functions1024

9.4 MacRobert's E-Function1025

9.41 Representation by means of multiple integrals1025

9.42 Functional relations1025

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 representations1026

9.52 Representation as a series or as an infinite product1026

9.53 Functional relations1027

9.54 Singular points and zeros1028

9.55 The Lerch functionφ（z,s,v）1028

9.56 The functionξ（s）1029

9.6 Bernoulli numbers and polynomials,Euler numbers1030

9.61 Bernoulli numbers1030

9.62 Bernoulli polynomials1031

9.63 Euler numbers1032

9.64 The functions v（x），v（x.α），μ（x，β），μ（x，β，α）．λ（x，y）1033

9.65 Euler polynomials1033

9.7 Constants1035

9.71 Bernoulli numbers1035

9.72 Euler numbers1035

9.73 Euler's and Catalan's constants1036

9.74 Stirling numbers1036

10 Vector Field Theory1039

10.1-10.8 Vectors,Vector Operators,and Integral Theorems1039

10.11 Products of vectors1039

10.12 Properties of scalar product1039

10.13 Properties of vector product1039

10.14 Differentiation of vectors1039

10.21 Operators grad,div,and curl1040

10.31 Properties of the operator ？1040

10.41 Solenoidal fields1041

10.51-10.61 Orthogonal curvilinear coordinates1042

10.71-10.72 Vector integral theorems1045

10.81 Integral rate of change theorems1047

11 Algebraic Inequalities1049

11.1-11.3 General Algebraic Inequalities1049

11.11 Algebraic inequalities involving real numbers1049

11.21 Algebraic inequalities involving complex numbers1050

11.31 Inequalities for sets of complex numbers1051

12 Integral Inequalities1053

12.11 Mean value theorems1053

12.111 First mean value theorem1053

12.112 Second mean value theorem1053

12.113 First mean value theorem for infinite integrals1053

12.114 Second mean value theorem for infinite integrals1054

12.21 Differentiation of definite integral containing a parameter1054

12.211 Differentiation when limits are finite1054

12.212 Differentiation when a limit is infinite1054

12.31 Integral inequalities1054

12.311 Cauch-Schwarz-Buniakowsky inequality for integrals1054

12.312 H？lder's inequality for integrals1054

12.313 Minkowski's inequality for integrals1055

12.314 Chebyshev's inequality for integrals1055

12.315 Young's inequality for integrals1055

12.316 Steffensen's inequality for integrals1055

12.317 Grams inequality for integrals1055

12.318 Ostrowski's inequality for integrals1055

12.41 Convexity and Jensen's inequality1056

12.411 Jensen's inequality1056

12.51 Fourier series and related inequalities1056

12.511 Riemann-Lebesgue lemma1056

12.512 Dirichlet lemma1057

12.513 Parseval's theorem for trigonometric Fourier series1057

12.514 Integral representation of the nth partial sum1057

12.515 Generalized Fourier series1057

12.516 Bessel's inequality for generalized Fourier series1057

12.517 Parseval's theorem for generalized Fourier series1057

13 Matrices and related results1059

13.11-13.12 Special matrices1059

13.111 Diagonal matrix1059

13.112 Identity matrix and null matrix1059

13.113 Reducible and irreducible matrices1059

13.114 Equivalent matrices1059

13.115 Transpose of a matrix1059

13.116 Adjoint matrix1059

13.117 Inverse matrix1060

13.118 Trace of a matrix1060

13.119 Symmetric matrix1060

13.120 Skew-symmetric matrix1060

13.121 Triangular matrices1060

13.122 Orthogonal matrices1060

13.123 Hermitian transpose of a matrix1060

13.124 Hermitian matrix1060

13.125 Unitary matrix1060

13.126 Eigenvalues and eigenvectors1061

13.127 Nilpotent matrix1061

13.128 Idempotent matrix1061

13.129 Positive definite1061

13.130 Non-negative definite1061

13.131 Diagonally dominant1061

13.21 Quadratic forms1061

13.211 Sylvester's law of inertia1062

13.212 Rank1062

13.213 Signature1062

13.214 Positive definite and semidefinite quadratic form1062

13.215 Basic theorems on quadratic forms1062

13.31 Differentiation of matrices1063

13.41 The matrix exponential1064

3.411 Basic properties1064

14 Determinants1065

14.11 Expansion of second- and third-order determinants1065

14.12 Basic properties1065

14.13 Minors and cofactors of a determinant1065

14.14 Principal minors1066

14.15 Laplace expansion of a determinant1066

14.16 Jacobi's theorem1066

14.17 Hadamard's theorem1066

14.18 Hadamard's inequality1067

14.21 Cramer's rule1067

14.31 Some special determinants1068

14.311 Vandermonde's determinant（alternant）1068

14.312 Circulants1068

14.313 Jacobian determinant1068

14.314 Hessian determinants1069

14.315 Wronskian determinants1069

14.316 Properties1069

14.317 Gram-Kowalewski theorem on linear dependence1070

15 Norms1071

15.1-15.9 Vector Norms1071

15.11 General properties1071

15.21 Principal vector norms1071

15.211 The norm ‖x‖11071

15.212 The norm ‖x‖2（Euclidean or L2 norm）1071

15.213 The norm ‖x‖∞1071

15.31 Matrix norms1072

15.311 General properties1072

15.312 Induced norms1072

15.313 Natural norm of unit matrix1072

15.41 Principal natural norms1072

15.411 Maximum absolute column sum norm1072

15.412 Spectral norm1072

15.413 Maximum absolute row sum norm1072

15.51 Spectral radius of a square matrix1073

15.511 Inequalities concerning matrix norms and the spectral radius1073

15.512 Deductions from Gerschgorin's theorem（see 15.814）1073

15.61 Inequalities involving eigenvalues of matrices1074

15.611 Cayley-Hamilton theorem1074

15.612 Corollaries1074

15.71 Inequalities for the characteristic polynomial1074

15.711 Named and unnamed inequalities1075

15.712 Parodi's theorem1076

15.713 Corollary of Brauer's theorem1076

15.714 Ballieus theorem1076

15.715 Routh-Hurwitz theorem1076

15.81-15.82 Named theorems on eigenvalues1076

15.811 Schur's inequalities1077

15.812 Sturmian separation theorem1077

15.813 Poincare's separation theorem1077

15.814 Gerschgorin's theorem1078

15.815 Brauers theorem1078

15.816 Perron's theorem1078

15.817 Frobenius theorem1078

15.818 Perron-Frobenius theorem1078

15.819 Wielandt's theorem1078

15.820 Ostrowskis theorem1079

15.821 First theorem due to Lyapunov1079

15.822 Second theorem due to Lyapunov1079

15.823 Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov1079

15.91 Variational principles1081

15.911 Rayleigh quotient1081

15.912 Basic theorems1081

16 Ordinary differential equations1083

16.1-16.9 Results relating to the solution of ordinary differential equations1083

16.11 First-order equations1083

16.111 Solution of a first-order equation1083

16.112 Cauchy problem1083

16.113 Approximate solution to an equation1083

16.114 Lipschitz continuity of a function1084

16.21 Fundamental inequalities and related results1084

16.211 Gronwall's lemma1084

16.212 Comparison of approximate solutions of a differential equation1084

16.31 Firstorder systems1085

16.311 Solution of a system of equations1085

16.312 Cauchy problem for a system1085

16.313 Approximate solution to a system1085

16.314 Lipschitz continuity of a vector1085

16.315 Comparison of approximate solutions of a system1086

16.316 First-order linear differential equation1086

16.317 Linear systems of differential equations1086

16.41 Some special types of elementary differential equations1087

16.411 Variables separable1087

16.412 Exact differential equations1087

16.413 Conditions for an exact equation1087

16.414 Homogeneous differential equations1087

16.51 Second-order equations1088

16.511 Adjoint and self-adjoint equations1088

16.512 Abel's identity1088

16.513 Lagrange identity1089

16.514 The Riccati equation1089

16.515 Solutions of the Riccati equation1089

16.516 Solution of a second-orderlinear differential equation1090

16.61-16.62 Oscillation and non-oscillation theorems for secondorder equations1090

16.611 First basic comparison theorem1090

16.622 Second basic comparison theorem1091

16.623 Interlacing of zeros1091

16.624 Sturm separation theorem1091

16.625 Sturm comparison theorem1091

16.626 Szeg？'s comparison theorem1091

16.627 Picone's identity1092

16.628 Sturm-Picone theorem1092

16.629 Oscillation on the half line1092

16.71 Two related comparison theorems1093

16.711 Theorem11093

16.712 Theorem21093

16.81-16.82 Non-oscillatory solutions1093

16.811 Kneser's non-oscillation theorem1094

16.822 Comparison theorem for non-oscillation1094

16.823 Necessary and sufficient conditions for non-oscillation1094

16.91 Some growth estimates for solutions of second-order equations1094

16.911 Strictly increasing and decreasing solutions1094

16.912 General result on dominant and subdominant solutions1095

16.913 Estimate of dominant solution1095

16.914 A theorem due to Lyapunov1096

16.92 Boundedness theorems1096

16.921 All solutions of the equation1096

16.922 If all solutions of the equation1096

16.923 If a（x）→∞monotonically as x→∞，then all solutions of1096

16.924 Consider the equation1096

16.93 Growth of maxima of｜y｜1097

17 Fourier,Laplace,and Mellin Transforms1099

17.1-17.4 Integral Transforms1099

17.11 Laplace transform1099

17.12 Basic properties of the Laplace transform1099

17.13 Table of Laplace transform pairs1100

17.21 Fourier transform1109

17.22 Basic properties of the Fourier transform1110

17.23 Table of Fourier transform pairs1110

17.24 Table of Fourier transform pairs for spherically symmetric functions1112

17.31 Fourier sine and cosine transforms1113

17.32 Basic properties of the Fourier sine and cosine transforms1113

17.33 Table of Fourier sine transforms1114

17.34 Table of Fourier cosine transforms1118

17.35 Relationships between transforms1121

17.41 Mellin transform1121

17.42 Basic properties of the Mellin transform1122

17.43 Table of Mellin cosine transforms1122

18 The z-transform1127

18.1-18.3 Definition,Bilateral,and Unilateral z-Transforms1127

18.1 Definitions1127

18.2 Bilateral z-transform1127

18.3 Unilateral z-transform1129

References1133

Supplemental references1137

Function and constant index1143

General index1153