Table of Integrals, Series, and Products

Preface to the Corrected and Enlarged Edition

Preface to the First Edition

Preface to the Third Edition

Preface to the Fourth Edition

Editor's Foreword

Acknowledgment

The Order of Presentation of the Formulas

Use of the Tables

Index of Special Functions and Notations

Notations

Note on the Bibliographic References


0. Introduction


0.1 FInite Sums


0.11 Progressions


0.12 Sums of Powers of Natural Numbers


0.13 Sums of Reciprocals of Natural Numbers


0.14 Sums of Products of Reciprocals of Natural Numbers


0.15 Sums of the BInomial Coefficients


0.2 Numerical Series and Infinite Products


0.21 The Convergence of Numerical Series


0.22 Convergence Tests


0.23-0.24 Examples of Numerical Series


0.25 InfInite Products


0.26 Examples of Infinite Products


0.3 Functional Series


0.30 DefInitions and Theorems


0.31 Power Series


0.32 Fourier Series


0.33 Asymptotic Series


0.4 CertaIn Formulas From Differential Calculus


0.41 Differentiation of a Definite Integral with Respect to a Parameter


0.42 The nth Derivative of a Product


0.43 The nth Derivative of a Composite Function


0.44 Integration by Substitution


1. Elementary Functions


1.1 Power of Binomials


1.11 Power Series


1.12 Series of Rational Fractions


1.2 The Exponential Function


1.21 Series Representations


1.22 Functional Relations


1.23 Series of Exponentials


1.3-1.4 Trigonometric and Hyperbolic Functions


1.30 Introduction


1.31 The Basic Functional Relations


1.32 The Representation of Powers of Trigonometric and Hyperbolic Functions in Terms of Functions of Multiples of the Argument (Angle)


1.33 The Representation of Trigonometric and Hyperbolic Functions of Multiples of the Argument (angle) in Terms of Powers of these Functions


1.34 Certain Sums of Trigonometric and Hyperbolic Functions


1.35 Sums of Powers of Trigonometric Functions of Multiple Angles


1.36 Sums of Products of Trigonometric Functions of Multiple Angles


1.37 Sums of Tangents of Multiple Angles


1.38 Sums Leading To Hyperbolic Tangents and Cotangents


1.39 The Representation of Cosines and Sines of Multiples of the Angle as Finite Products


1.41 The Expansion of Trigonometric and Hyperbolic Functions in Power Series


1.42 Expansion in Series of Simple Fractions


1.43 Representation in the Form of an Infinite Product


1.44-1.45 Trigonometric (Fourier) Series


1.46 Series of Products of Exponential and Trigonometric Functions


1.47 Series of Hyperbolic Functions


1.48 Lobachevskiy's "Angle of Parallelism" Π (x)


1.49 The Hyperbolic Amplitude (The Gudermannian) gd x


1.5 The Logarithm


1.51 Series Representation


1.52 Series of Logarithms


1.6 The Inverse Trigonometric and Hyperbolic Functions


1.61 The Domain of Definition


1.62-1.63 Functional Relations


2. Indefinite Integals of Elementary Functions


2.0 Introduction


2.00 General Remarks


2.01 The Basic Integrals


2.02 General Formulas


2.1 Rational Functions


2.10 General Integration Rules


2.11-2.13 Forms Containing the Binomial a + bxk


2.14 Forms Containing the Binomial 1 ± xn


2.15 Forms Containing Pairs of Binomials: a + bx and α + βx


2.16 Forms Containing the Trinomial a + bxk + cx2k


2.17 Forms Containing the Quadratic Trinomial a + bx + Cx2 and Powers of x


2.18 Forms Containing the Quadratic Trinomial a + bx + Cx2 and the Binomial α + βx


2.2 Algebraic Functions


2.20 Introduction


2.21 Forms Containing the Binomial a + bxk and √x


2.22-2.23 Forms Containing x√(a + bx)k


2.24 Forms Containing √a + bx and he Binomial α + βx


2.25 Forms Containing √a + bx + cx2


2.26 Forms Containing √a + bx + cx2 and Integral Powers of x


2.27 Forms Containing √a + cx2 and Integral Powers of x


2.28 Forms Containing √a + bx + cx2 and First- and Second-Degree Polynomials


2.29 Integrals that can be Reduced o Elliptic Or Pseudo-Elliptic Integrals


2.3 The Exponential Function


2.31 Forms Containing eax


2.32 The Exponential Combined with Rational Functions of x


2.4 Hyperbolic Functions


2.41-2.43 Powers of Sh x, ch x, th x and cth x


2.44-2.45 Rational Functions of Hyperbolic Functions


2.46 Algebraic Functions of Hyperbolic Functions


2.47 Combinations of Hyperbolic Functions and Powers


2.48 Combinations of Hyperbolic Functions, Exponentials, and Powers


2.5-2.6 Trigonometric Functions


2.50 Introduction


2.51-2.52 Powers of Trigonometric Functions


2.53-2.54 Sines and Cosines of Multiple Angles and of Linear and More Complicated Functions of the Argument


2.55-2.56 Rational Functions of the Sine and Cosine


2.57 Forms Containing √a ± b sin x, Or √a ± b cos x and Forms Reducible to Such Expressions


2.58-2.62 Integrals Reducible to Elliptic and Pseudo-Elliptic Integrals


2.63-2.65 Products of Trigonometric Functions and Powers


2.66 Combinations of Trigonometric Functions and Exponentials


2.67 Combinations of Trigonometric and Hyperbolic Functions


2.7 Logarithms and Inverse-Hyperbolic Functions


2.71 The Logarithm


2.72-2.73 Combinations of Logarithms and Algebraic Functions


2.74 Inverse Hyperbolic Functions


2.8 Inverse Trigonometric Functions


2.81 Arcsines and Arccosines


2.82 The Arcsecant, the Arccosecant, the Arctangent, and the Arccotangent


2.83 Combinations of Arcsine or Arccosine and Algebraic Functions


2.84 Combinations of the Arcsecant and Arccosecant with Powers of x


2.85 Combinations of the Arctangent and Arccotangent with Algebraic Functions


3.-4. Definite Integrals of Elementary Functions


3.0 Introduction


3.01 Theorems of A General Nature


3.02 Change of Variable in A Definite Integral


3.03 General Formulas


3.04 Improper Integrals


3.05 The Principal Values of Improper Integrals


3.1-3.2 Powerand Algebraic Functions


3.11 Rational Functions


3.12 Products of Rational Functions and Expressions that can be Reduced to Square Roots of First- and Second-Degree Polynomials


3.13-3.17 Expressions that can be Reduced to Square Roots of Third- and Fourth-Degree Polynomials and their Products with Rational Functions


3.18 Expressions that can be Reduced to Fourth Roots of Second-Degree Polynomials and their Products with Rational Functions


3.19-3.23 Combinations of Powers of x and Powers of Binomials of the Form (a + βx)


3.24-3.27 Powers of x, of Binomials of he Form a + βxp and of Polynomials in x


3.3-3.4 Exponential Functions


3.31 Exponential Functions


3.32-3.34 Exponentials of More Complicated Arguments


3.35 Combinations of Exponentials and Rational Functions


3.36-3.37 Combinations of Exponentials and Algebraic Functions


3.38-3.39 Combinations of Exponentials and Arbitrary Powers


3.41-3.44 Combinations of Rational Functions of Powers and Exponentials


3.45 Combinations of Powers and Algebraic Functions of Exponentials


3.46-3.48 Combinations of Exponentials of More Complicated Arguments and Powers


3.5 Hyperbolic Functions


3.51 Hyperbolic Functions


3.52-3.53 Combinations of Hyperbolic Functions and Algebraic Functions


3.54 Combinations of Hyperbolic Functions and Exponentials


3.55-3.56 Combinations of Hyperbolic Functions, Exponentials, and Powers


3.6-4.1 Trigonometric Functions


3.61 Rational Functions of Sines and Cosines and Trigonometric Functions of Multiple Angles


3.62 Powers of Trigonometric Functions


3.63 Powers of Trigonometric Functions and Trigonometric Functions of Linear Functions


3.64-3.66 Powers and Rational Functions of Trigonometric Functions


3.67 Square Roots of Expressions Containing Trigonometric Functions


3.68 Various Forms of Powers of Trigonometric Functions


3.69-3.71 Trigonometric Functions of More Complicated Arguments


3.72-3.74 Combinations of Trigonometric and Rational Functions


3.75 Combinations of Trigonometric and Algebraic Functions


3.76-3.77 Combinations of Trigonometric Functions and Powers


3.78-3.81 Rational Functions of x and of Trigonometric Functions


3.82-3.83 Powers of Trigonometric Functions Combined With Other Powers


3.84 Integrals Containing the Expressions √1 - k2 sin2 x, √1 - k2 cos2 x, and Similar Expressions


3.85-3.88 Trigonometric Functions of More Complicated Arguments Combined With Powers


3.89-3.91 Trigonometric Functions and Exponentials


3.92 Trigonometric Functions of More Complicated Arguments CombIned with Exponentials


3.93 Trigonometric and Exponential Functions of Trigonometric Functions


3.94-3.97 Combinations Involving Trigonometric Functions, Exponentials, and Powers


3.98-3.99 Combinations of Trigonometric and Hyperbolic Functions


4.11-4.12 Combinations Involving Trigonometric and Hyperbolic Functions and Powers


4.13 Combinations of Trigonometric and Hyperbolic Functions and Exponentials


4.14 Combinations of Trigonometric and Hyperbolic Functions, Exponentials, and Powers


4.2-4.4 Logarithmic Functions


4.21 Logarithmic Functions


4.22 Logarithms of More Complicated Arguments


4.23 Combinations of Logarithms and Rational Functions


4.24 Combinations of Logarithms and Algebraic Functions


4.25 Combinations of Logarithms and Powers


4.26-4.27 Combinations Involving Powers of the Logarithm and Other Powers


4.28 Combinations of Rational Functions of in Jc and Powers


4.29-4.32 Combinations of Logarithmic Functions of More Complicated Arguments and Powers


4.33-4.34 Combinations of Logarithms and Exponentials


4.35-4.36 Combinations of Logarithms, Exponentials, and Powers


4.37 Combinations of Logarithms and Hyperbolic Functions


4.38-4.41 Logarithms and Trigonometric Functions


4.42-4.43 Combinations of Logarithms, Trigonometric Functions, and Powers


4.44 Combinations of Logarithms, Trigonometric Functions, and Exponentials


4.5 Inverse Trigonometric Functions


4.51 Inverse Trigonometric Functions


4.52 Combinations of Arcsines, Arccosines, and Powers


4.53-4.54 Combinations of Arctangents, Arccotangents, and Powers


4.55 Combinations of Inverse Trigonometric Functions and Exponentials


4.56 A Combination of The Arctangent and A Hyperbolic Function


4.57 Combinations of Inverse and Direct Trigonometric Functions


4.58 A Combination InvolvIng an Inverse and A Direct Trigonometric Function and a Power


4.59 Combinations of Inverse Trigonometric Functions and Logarithms


4.6 Multiple Integrals


4.60 Change of Variables in Multiple Integrals


4.61 Change of he Order of Integration and Change of Variables


4.62 Double and Triple Integrals With Constant Limits


4.63-4.64 Multiple Integrals


5. Indefinite Integrals of Special Functions


5.1 Elliptic Integrals and Functions


5.11 Complete Elliptic Integrals


5.12 Elliptic Integrals


5.13 Jacobian Elliptic Functions


5.14 Weierstrass Elliptic Functions


5.2 The Exponential-Integral Function


5.21 The Exponential-Integral Function


5.22 Combinations of the Exponential-Integral Function and Powers


5.23 Combinations of the Exponential-Integral and the Exponential


5.3 The Sine-Integral and the Cosine-Integral


5.4 The Probability Integral and Fresnel Integrals


5.5 Bessel Functions


6.-7. DefInite Integrals of Special Functions


6.1 Elliptic Integrals and Functions


6.11 Forms Containing F (x, k)


6.12 Forms Containing E (x, k)


6.13 Integration of Elliptic Integrals With Respect to the Modulus


6.14-6.15 Complete Elliptic Integrals


6.16 The Theta Function


6.2-6.3 The Exponential-Integral Function and Functions Generated By It


6.21 The Logarithm-Integral


6.22-6.23 The Exponential-Integral Function


6.24-6.26 The Sine- and Cosine-Integral Functions


6.27 The Hyperbolic-Sine- and Cosine-Integral Functions


6.28-6.31 The Probability Integral


6.32 Fresnel Integrals


6.4 The Gamma Function and Functions Generated By It


6.41 The Gamma Function


6.42 Combinations of the Gamma Function, the Exponential, and Powers


6.43 Combinations of the Gamma Function and Trigonometric Functions


6.44 The Logarithm of the Gamma Function


6.45 The Incomplete Gamma Function


6.46-6.47 The Function ψ (x)


6.5-6.7 Bessel Functions


6.51 Bessel Functions


6.52 Bessel Functions Combined with x and x2


6.53-6.54 Combinations of Bessel Functions and Rational Functions


6.55 Combinations of Bessel Functions and Algebraic Functions


6.56-6.58 Combinations of Bessel Functions and Powers


6.59 Combinations of Powers and Bessel Functions of More Complicated Arguments


6.61 Combinations of Bessel Functions and Exponentials


6.62-6.63 Combinations of Bessel Functions, Exponentials, and Powers


6.64 Combinations of Bessel Functions of More Complicated Arguments, Exponentials, and Powers


6.65 Combinations of Bessel and Exponential Functions of More Complicated Arguments and Powers


6.66 Combinations of Bessel, Hyperbolic, and Exponential Functions


6.67-6.68 Combinations of Bessel and Trigonometric Functions


6.69-6.74 Combinations of Bessel and Trigonometric Functions and Powers


6.75 Combinations of Bessel, Trigonometric, and Exponential Functions and Powers


6.76 Combinations of Bessel, Trigonometric, and Hyperbolic Functions


6.77 Combinations of Bessel Functions and the Logarithm, or Arctangent


6.78 Combinations of Bessel and Other Special Functions


6.79 Integration of Bessel Functions with Respect to the Order


6.8 Functions Generated By Bessel Functions


6.81 Struve Functions


6.82 Combinations of Struve Functions, Exponentials, and Powers


6.83 Combinations of Struve and Trigonometric Functions


6.84-6.85 Combinations of Struve and Bessel Functions


6.86 Lommel Functions


6.87 Thomson Functions


6.9 Mathieu Functions


6.91 Mathieu Functions


6.92 Combinations of Mathieu, Hyperbolic, and Trigonometric Functions


6.93 Combinations of Mathieu and Bessel Functions


7.1-7.2 Associated Legendre Functions


7.11 Associated Legendre Functions


7.12-7.13 Combinations of Associated Legendre Functions and Powers


7.14 Combinations of Associated Legendre Functions, Exponentials, and Powers


7.15 Combinations of Associated Legendre and Hyperbolic Functions


7.16 Combinations of Associated Legendre Functions, Powers, and Trigonometric Functions


7.17 A Combination of an Associated Legendre Function and the Probability Integral


7.18 Combinations of Associated Legendre and Bessel Functions


7.19 Combinations of Associated Legendre Functions and Functions Generated By Bessel Functions


7.21 Integration of Associated Legendre Functions with Respect to the Order


7.22 Combinations of Legendre Polynomials, Rational Functions, and Algebraic Functions


7.23 Combinations of Legendre Polynomials and Powers


7.24 Combinations of Legendre Polynomials and Other Elementary Functions


7.25 Combinations of Legendre Polynomials and Bessel Functions


7.3-7.4 Orthogonal Polynomials


7.31 Combinations of Gegenbauer Polynomials Cvn (x) and Powers


7.32 Combinations of the Polynomials Cvn (x) and Some Elementary Functions


7.33 Combinations of the Polynomials Cvn (x) and Bessel Functions. Integration of Gegenbauer Functions with Respect to the Index


7.34 Combinations of Chebyshev Polynomials and Powers


7.35 Combinations of Chebyshev Polynomials and Some Elementary Functions


7.36 Combinations of Chebyshev Polynomials and Bessel Functions


7.37-7.38 Hermite Polynomials


7.39 Jacobi Polynomials


7.41-7.42 Laguerre Polynomials


7.5 Hypergeometric Functions


7.51 Combinations of Hypergeometric Functions and Powers


7.52 Combinations of Hypergeometric Functions and Exponentials


7.53 Hypergeometric and Trigonometric Functions


7.54 Combinations of Hypergeometric and Bessel Functions


7.6 Degenerate Hypergeometric Functions


7.61 Combinations of Degenerate Hypergeometric Functions and Powers


7.62-7.63 Combinations of Degenerate Hypergeometric Functions and Exponentials


7.64 Combinations of Degenerate Hypergeometric and Trigonometric Functions


7.65 Combinations of Degenerate Hypergeometric Functions and Bessel Functions


7.66 Combinations of Degenerate Hypergeometric Functions, Bessel Functions, and Powers


7.67 Combinations of Degenerate Hypergeometric Functions, Bessel Functions, Exponentials, and Powers


7.68 Combinations of Degenerate Hypergeometric Functions and Other Special Functions


7.69 Integration of Degenerate Hypergeometric Functions With Respect to the Index


7.7 Parabolic-Cylinder Functions


7.71 Parabolic-Cylinder Functions


7.72 Combinations of Parabolic-Cylinder Functions, Powers, and Exponentials


7.73 Combinations of Parabolic-Cylinder and Hyperbolic Functions


7.74 Combinations of Parabolic-Cylinder and Trigonometric Functions


7.75 Combinations of Parabolic-Cylinder and Bessel Functions


7.76 Combinations of Parabolic-Cylinder Functions and Degenerate Hypergeometric Functions


7.77 Integration of A Parabolic-Cylinder Function with Respect to the Index


7.8 Meijer's and Macrobert's Functions (G and E)


7.81 Combinations of the Functions G and E and the Elementary Functions


7.82 Combinations of the Functions G and E and Bessel Functions


7.83 Combinations of the Functions G and E and Other Special Functions


8.-9. Special Functions


8.1 Elliptic Integrals and Functions


8.11 Elliptic Integrals


8.12 Functional Relations Between Elliptic Integrals


8.13 Elliptic Functions


8.14 Jacobian Elliptic Functions


8.15 Properties of Jacobian Elliptic Functions and Functional Relationships Between Them


8.16 The Weierstrass Function P (u)


8.17 The Functions ζ (u) and σ (u)


8.18-8.19 Theta Functions


8.2 The Exponential-Integral Function and Functions Generated By It


8.21 The Exponential-Integral Function Ei (x)


8.22 The Hyperbolic-Sine-Integral shi x and the Hyperbolic-Cosine-Integral chi x


8.23 The Sine Integral and the Cosine Integral: si (x) and ci (x)


8.24 The Logarithm-Integral li (x)


8.25 The Probability Integral and Fresnel Integrals φ (x), S (x), and C (x)


8.26 Lobachevskiy's Function L (x)


8.3 Euler's Integrals of the First and Second Kinds and Functions Generated By Them


8.31 The Gamma Function (Euler's Integral of the Second Kind): Γ (z)


8.32 Representation of the Gamma Function as Series and Products


8.33 Functional Relations Involving the Gamma Function


8.34 The Logarithm of the Gamma Function


8.35 The Incomplete Gamma Function


8.36 The Psi Function ψ (x)


8.37 The Function β(x)


8.38 The Beta Function (Euler's Integral of the First Kind): B (x, y)


8.4-8.5 Bessel Functions and Functions Associated with them


8.40 Definitions


8.41 Integral Representations of the Functions Jv (z) and Nv (z)


8.42 Integral Representations of the Functions H(1)v(z) and H (2)v (z)


8.43 Integral Representations of the Functions Iv (z) and Kv (z)


8.44 Series Representation


8.45 Asymptotic Expansions of Bessel Functions


8.46 Bessel Functions of Order Equal to an Integer Plus One-Half


8.47-8.48 Functional Relations


8.49 Differential Equations LeadIng to Bessel Functions


8.51-8.52 Series of Bessel Functions


8.53 Expansion in Products of Bessel Functions


8.54 The Zeros of Bessel Functions


8.55 Struve Functions


8.56 Thomson Functions and their Generalizations: ber, (z), bei, (z), her, (z), hei, (z), ker (z), kei (z)


8.57 Lommel Functions


8.58 Anger and Weber Functions Jv (z) and Ev (z)


8.59 Neumann's and Schläfiys Polynomials: On (z) and Sn (z)


8.6 Mathieu Functions


8.60 Mathieu's Equation


8.61 Periodic Mathieu Functions


8.62 Recursion Relations for the Coefficients A(2n)2r, A (2n+1)2r+1, B(2n+1)2r+1, B(2n+2)2r+2


8.63 Mathieu Functions with a Purely Imaginary Argument


8.64 Nonperiodic Solutions of Mathieu's Equation


8.65 Mathieu Functions For Negative q


8.66 Representation of Mathieu Functions as Series of Bessel Functions


8.67 The General Theory


8.7-8.8 Associated Legendre Functions


8.70 Introduction


8.72 Asymptotic Series for Large Values of