MATHEMATICAL FORMULAS AND INTEGRALS

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1 MATHEMATICAL FORMULAS AND INTEGRALS ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom Academic Press San Diego New York Boston London Sydney Tokyo Toronto

2 Preface xix Index of Special Functions and Notations xxi Q Quick Reference List of Frequently Used Data O.I Useful Identities Trigonometric identities Hyperbolic identities Complex Relationships Constants Derivatives of Elementary Functions Rules of Differentiation and Integration Standard Integrals Standard Series Geometry 13 Numerical, Algebraic, and Analytical Results for Series and Calculus 1.1 Algebraic Results Involving Real and Complex Numbers Complex numbers Algebraic inequalities involving real and complex numbers 26

3 vi 1.2 Finite Sums The binomial theorem for positive integral exponents Arithmetic, geometric, and arithmetic-geometric series Sums of powers of integers Bernoulli and Euler Numbers and Polynomials Bernoulli and Euler numbers Bernoulli and Euler polynomials The Euler-Maclaurin summation formula Determinants Expansion of second- and third-order determinants Minors, cofactors, and the Laplace expansion Basic properties of determinants Jacobi's theorem Hadamard's theorem Hadamard's inequality Cramer's rule Some special determinants Routh-Hurwitz theorem Matrices Special matrices Quadratic forms Differentiation and integration of matrices 57 : The matrix exponential The Gerschgorin circle theorem Permutations and Combinations Permutations Combinations Partial Fraction Decomposition Rational functions Method of undetermined coefficients Convergence of Series Types of convergence of numerical series Convergence tests Examples of infinite numerical series Infinite Products Convergence of infinite products Examples of infinite products Functional Series Uniform convergence Power Series Definition Taylor Series Definition and forms of remainder term Fourier Series Definitions 77

4 yjj 1.14 Asymptotic Expansions Introduction Definition and properties of asymptotic series Basic Results from the Calculus Rules for differentiation Integration Reduction formulas Improper integrals Integration of rational functions Elementary applications of definite integrals 91 Functions and Identities 2.1 Complex Numbers and Trigonometric and Hyperbolic Functions Basic results Logarithms and Exponentials Basic functional relationships The number e The Exponential Function Series representations Trigonometric Identities Trigonometric functions Hyperbolic Identities Hyperbolic functions The Logarithm Series representations Inverse Trigonometric and Hyperbolic Functions Domains of definition and principal values Functional relations Series Representations of Trigonometric and Hyperbolic Functions Trigonometric functions Hyperbolic functions Inverse trigonometric functions Inverse hyperbolic functions Useful Limiting Values and Inequalities Involving Elementary Functions Logarithmic functions Exponential functions Trigonometric and hyperbolic functions 133 Q Derivatives of Elementary Functions 3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions Derivatives of Trigonometric Functions 136

5 yjjj 3.3 Derivatives of Inverse Trigonometric Functions Derivatives of Hyperbolic Functions Derivatives of Inverse Hyperbolic Functions 138 A Indefinite Integrals of Algebraic Functions 4.1 Algebraic and Transcendental Functions Definitions Indefinite Integrals of Rational Functions Integrands involving x" Integrands involving a + bx Integrands involving linear factors Integrands involving a 2 ± b 2 x Integrands involving a + bx + ex Integrands involving a + bx Integrands involving a-)-fox Nonrational Algebraic Functions Integrands containing a + bx k and Jx Integrands containing (a + bx) l/ Integrands containing (a + ex 2 ) 1/ Integrands containing (a + bx + cx 2 ) l/2 160 J7 Indefinite Integrals of Exponential Functions 5.1 Basic Results Indefinite integrals involving e ax Integrands involving the exponential functions combined with rational functions of x Integrands involving the exponential functions combined with trigonometric functions 165 Indefinite Integrals of Logarithmic Functions 6.1 Combinations of Logarithms and Polynomials The logarithm Integrands involving combinations of In (ax) and powers of x Integrands involving (a + bx) m In"x Integrands involving ln(x 2 ± a 2 ) Integrands involving x m \n[x + (x 2 ± a 2 ) 1 ' 2 ] 174

6 JX 7 Indefinite Integrals of Hyperbolic Functions 7.1 Basic Results Integrands involving sinh(<z + bx) and cosh(a + bx) Integrands Involving Powers of sinh(fox) or cosh(fex) Integrands involving powers of sinh(fox) Integrands involving powers of cosh(fox) Integrands Involving (a ± bx) m sinh(cx) or (a + bx) m cosh(cx) General results Integrands Involving x m sinh"x or x m cosh"x Integrands involving x m sinh"x Integrals involving x m cosh"x Integrands Involving x m sinh~"x or x m cosh~"x Integrands involving x m sinh~"x Integrands involving x m cosh~"x Integrands Involving (1 ± cosh x)~ m Integrands involving (1 ± coshx)" Integrands involving (1 ± coshx)~ Integrands Involving sinh(ax)cosh~"x or cosh(ax)sinh~"x Integrands involving sinh(ax) cosh""x Integrands involving cosh(ax) sinh~"x Integrands Involving sinh(ax + b) and cosh(cx 4- d) General case Special case a c Integrands involving sinh p xcosh? x Integrands Involving tanh kx and coth kx Integrands involving tanh kx Integrands involving coth kx Integrands Involving (a + bx) m sinh kx or (a + bx) m cosh kx Integrands involving (a + bx) m sinh kx Integrands involving (a + bx) m cosh kx 185 Q Indefinite Integrals Involving Inverse Hyperbolic Functions 8.1 Basic Results Integrands involving products of x" and arcsinh(x/a) or arccosh(x/a) Integrands Involving x~" arcsinh(x/a) or x~" arccosh(x/a) Integrands involving x~ n arcsinh(x/a) Integrands involving x~" arccosh(x/a) Integrands Involving x" arctanh(x/a) or x" arccoth(x/a) Integrands involving x" arctanh(x/a) Integrands involving x" arccoth(x/a) 190

7 8.4 Integrands Involving x~" arctanh(x/a) or x~" arccoth(x/a) Integrands involving x~" arctanh(x/a) Integrands involving x~" arccoth(x/a) 191 Q Indefinite Integrals of Trigonometric Functions 9.1 Basic Results Simplification by means of substitutions Integrands Involving Powers of x and Powers of sin x or cos x Integrands involving x" sin m x Integrands involving x~ n sutx Integrands involving x" sin~ m x Integrands involving x" cos" 1 * Integrands involving x~"cos m x Integrands involving x" cos~ m x Integrands involving x" sinx/(a + bcosx) m or x"cosx/(a + fosinx) m Integrands Involving tan x and/or cot x Integrands involving tan" x or tan" x/(tanx ± 1) Integrands involving cot" x or tanx and cotx Integrands Involving sin x and cos x Integrands involving sin m x cos"x Integrands involving sin~"x Integrands involving cos~"x Integrands involving sin m x/cos"x or cos m x/sin"x Integrands involving sin~ m x cos~"x Integrands Involving Sines and Cosines with Linear Arguments and Powers ofx Integrands involving products of (ax + b)", sin (ex + a*), and/or cos(px + a) Integrands involving x" sin m x or x" cos m x 208 1Q Indefinite Integrals of Inverse Trigonometric Functions 10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions Integrands involving x" arcsin m (x/a) Integrands involving x~" arcsin(x/a) Integrands involving x" arccos"'(x/a) Integrands involving x~" arccos(x/a) Integrands involving x" arctan(x/a) Integrands involving x~" arctan(x/a) Integrands involving x" arccot(x/a) 214

8 Integrands involving x~" arccot(x/a) Integrands involving products of rational functions and arccot(x/a) 215 The Gamma, Beta, Pi, and Psi Functions 11.1 The Euler Integral and Limit and Infinite Product Representations fort(x) Definitions and notation Special properties of T(x) Asymptotic representations of F (x) and n! Special values of T(x) The gamma function in the complex plane The psi (digamma) function The beta function Graph of T(x) and tabular values of T(x) and In T(x) 221 J2 Elliptic Integrals and Functions 12.1 Elliptic Integrals Legendre normal forms Tabulations and trigonometric series representations of complete elliptic integrals Tabulations and trigonometric series for E{cp, k) and F(<p, k) Jacobian Elliptic Functions The functions sn u, en u, and dn u Basic results Derivatives and Integrals Derivatives of sn u, en u, and dn u Integrals involving sn u, en u, and dn u Inverse Jacobian Elliptic Functions Definitions 233 Probability Integrals and the Error Function 13.1 Normal Distribution Definitions Power series representations (x > 0) Asymptotic expansions (x» 0) The Error Function Definitions 238

9 Xii Power series representation Asymptotic expansion (x» 0) Connection between P(x) and erf x Integrals expressible in terms of erf x Derivatives of erf x Integrals of erfcx Integral and power series representation of i" erfc x Value of i" erfc x at zero 240 Fresnel Integrals 14.1 Definitions, Series Representations, and Values at Infinity The Fresnel integrals Series representations Limiting values as x > oo 242 IK Definite Integrals 15.1 Integrands Involving Powers of x Integrands Involving Trigonometric Functions Integrands Involving the Exponential Function Integrands Involving the Hyperbolic Function Integrands Involving the Logarithmic Function Different Forms of Fourier Series 16.1 Fourier Series for f(x) on n < x < n The Fourier series Fourier Series for f(x) on L < x < L The Fourier series Fourier Series for /(x) on a < x < b The Fourier series Half-Range Fourier Cosine Series for /(x) on 0 < x < n The Fourier series Half-Range Fourier Cosine Series for /(x) on 0 < x < L The Fourier series Half-Range Fourier Sine Series for f(x) on 0 < x < n The Fourier series Half-Range Fourier Sine Series for f(x) on 0 < x < L The Fourier series 256

10 16.8 Complex (Exponential) Fourier Series for f(x) on n < x < n The Fourier series Complex (Exponential) Fourier Series for /( x) on L < x < L The Fourier series Representative Examples of Fourier Series Fourier Series and Discontinuous Functions Periodic extensions and convergence of Fourier series Applications to closed-form summations of numerical series Bessel Functions 17.1 Bessel's Differential Equation Different forms of Bessel's equation Series Expansions for / u (x) and Y v {x) Series expansions for J n (x) and J v (x) Series expansions for Y n {x) and Y v (x) Bessel Functions of Fractional Order Bessel functions 7± (n+ 1/2) (x) Bessel functions y ±(n+ i/2)(x) Asymptotic Representations of Bessel Functions Asymptotic representations for large arguments Asymptotic representations for large orders Zeros of Bessel Functions Zeros of J n (x) and Y n (x) Bessel's Modified Equation Different forms of Bessel's modified equation Series Expansions for I v (x) and K v (x) Series expansions for I n (x) and / v (x) Series expansions for K 0 (x) and K n (x) Modified Bessel Functions of Fractional Order Modified Bessel functions/ ±(n+ i/ 2 )(x) Modified Bessel functions AT ±(n+ i/2)(x) Asymptotic Representations of Modified Bessel Functions Asymptotic representations for large arguments Relationships between Bessel Functions Relationships involving J v (x) and Y v (x) Relationships involving / v (x) and K v (x) Integral Representations of J n (x), l n {x), and K n (x) Integral representations of J n (x) Indefinite Integrals of Bessel Functions Integrals of 7 n (x),/ n (x), and K n (x) Definite Integrals Involving Bessel Functions Definite integrals involving J n (x) and elementary functions 279

11 1Q Orthogonal Polynomials 18.1 Introduction Definition of a system of orthogonal polynomials Legendre Polynomials P n (x) Differential equation satisfied by P n (x) Rodrigue's formula for P n (x) Orthogonality relation for P n (x) Explicit expressions for P n (x) Recurrence relations satisfied by P n (x) Generating function for P n (x) Legendre functions of the second kind Q n {x) Chebyshev Polynomials T n (x) and U n (x) Differential equation satisfied by T n (x) and U n (x) Rodrigue's formulas for T n (x) and U n (x) Orthogonality relations for T n (x) and U n (x) Explicit expressions for T n (x) and U n (x) Recurrence relations satisfied by T n (x) and U n (x) Generating functions for T n (x) and U n (x) Laguerre Polynomials L n (x) Differential equation satisfied by L n (x) Rodrigue's formula for L n (x) Orthogonality relation for L n (x) Explicit expressions for L n (x) Recurrence relations satisfied by L n (x) Generating function for L n {x) Hermite Polynomials H n {x) Differential equation satisfied by H n (x) Rodrigue's formula for H n (x) Orthogonality relation for H n (x) Explicit expressions for H n (x) Recurrence relations satisfied by H n (x) Generating function for H n (x) 292 Laplace Transformation 19.1 Introduction Definition of the Laplace transform Basic properties of the Laplace transform The Dirac delta function S(x) Laplace transform pairs 295

12 20 Fourier Transforms 20.1 Introduction Fourier exponential transform Basic properties of the Fourier transforms Fourier transform pairs Fourier cosine and sine transforms Basic properties of the Fourier cosine and sine transforms Fourier cosine and sine transform pairs 306 XV Numerical Integration 21.1 Classical Methods Open- and closed-type formulas Composite midpoint rule (open type) Composite trapezoidal rule (closed type) Composite Simpson's rule (closed type) Newton-Cotes formulas Gaussian quadrature (open type) Romberg integration (closed type) Solutions of Standard Ordinary Differential Equations 22.1 Introduction Basic definitions Linear dependence and independence Separation of Variables Linear First-Order Equations Bernoulli's Equation Exact Equations Homogeneous Equations Linear Differential Equations Constant Coefficient Linear Differential Equations Homogeneous Case Linear Homogeneous Second-Order Equation Constant Coefficient Linear Differential Equations Inhomogeneous Case Linear Inhomogeneous Second-Order Equation Determination of Particular Integrals by the Method of Undetermined Coefficients The Cauchy-Euler Equation Legendre's Equation 331

13 22.15 Bessel's Equations Power Series and Frobenius Methods The Hypergeometric Equation Numerical Methods 339 Vector Analysis 23.1 Scalars and Vectors Basic definitions Vector addition and subtraction Scaling vectors Vectors in component form Scalar Products Vector Products Triple Products Products of Four Vectors Derivatives of Vector Functions of a Scalar t Derivatives of Vector Functions of Several Scalar Variables Integrals of Vector Functions of a Scalar Variable / Line Integrals Vector Integral Theorems A Vector Rate of Change Theorem Useful Vector Identities and Results 362 Systems of Orthogonal Coordinates 24.1 Curvilinear Coordinates Basic definitions Vector Operators in Orthogonal Coordinates Systems of Orthogonal Coordinates 365 2$ Partial Differential Equations and Special Functions 25.1 Fundamental Ideas Classification of equations Method of Separation of Variables Application to a hyperbolic problem The Sturm-Liouville Problem and Special Functions A First-Order System and the Wave Equation Conservation Equations (Laws) The Method of Characteristics 386

14 XVM 25.7 Discontinuous Solutions (Shocks) Similarity Solutions Burgers's Equation, the KdV Equation, and the KdVB Equation 394 Short Classified Reference List 397 Index 401

MATHEMATICAL FORMULAS AND INTEGRALS

HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS A Harcourt