H A N D B O O K OF

MATHEMATICALFORMULASAND INTEGRALSSecond Edition

ALAN JEFFREYDepartment of Engineering MathematicsUniversity of Newcastle upon TyneNewcastle upon TyneUnited Kingdom

ACADEMIC PRESSA Harcourt Science and Technology Company

San Diego San Francisco New York Boston London Sydney Tokyo

H A N D B O O K OF

MATHEMATICALFORMULASAMD INTEGRALSSecond Edition

ALAN JEFFREYDepartment of Engineering MathematicsUniversity of Newcastle upon TyneNewcastle upon TyneUnited Kingdom

ACADEMIC PRESSA Harcourt Science and Technology Company

San Diego San Francisco New York Boston London Sydney Tokyo

Contents

Preface xixPreface to the Second Edition xxiIndex of Special Functions and Notations xxiii

Q Quick Reference List of Frequently Used Data

0.1 Useful Identities 10.1.1 Trigonometric identities 1

_ 0.1.2 Hyperbolic identities 20.2 Complex Relationships 20.3 Constants 20.4 Derivatives of Elementary Functions 30.5 Rules of Differentiation and Integration 3 ,0.6 Standard Integrals 40.7 Standard Series 110.8 Geometry 13

Numerical, Algebraic, and Analytical Results for Seriesand Calculus

1.1 Algebraic Results Involving Real and Complex Numbers 251.1.1 Complex numbers 251.1.2 Algebraic inequalities involving real and complex numbers 26

Contents

1.2 Finite Sums 291.2.1 The binomial theorem for positive integral

exponents 291.2.2 Arithmetic, geometric, and arithmetic-geometric

series 331.2.3 Sums of powers of integers 341.2.4 Proof by mathematical induction 36

1.3 Bernoulli and Euler Numbers and Polynomials 371.3.1 Bernoulli and Euler numbers 371.3.2 Bernoulli and Euler polynomials 431.3.3 The Euler-Maclaurin summation formula 451.3.4 Accelerating the convergence of alternating series 46

1.4 Determinants 471.4.1 Expansion of second- and third-order determinants 471.4.2 Minors, cofactors, and the Laplace expansion 481.4.3 Basic properties of determinants 501.4.4 Jacobi's theorem 501.4.5 Hadamard's theorem 511.4.6 Hadamard's inequality 511.4.7 Cramer's rule 521.4.8 Some special determinants 521.4.9 Routh-Hurwitz theorem 54

1.5 Matrices 551.5.1 Special matrices 551.5.2 Quadratic forms 581.5.3 Differentiation and integration of matrices 601.5.4 The matrix exponential 611.5.5 The Gerschgorin circle theorem 61

1.6 Permutations and Combinations 621.6.1 Permutations 621.6.2 Combinations 62

1.7 Partial Fraction Decomposition 631.7.1 Rational functions 631.7.2 Method of undetermined coefficients 63

1.8 Convergence of Series 661.8.1 Types of convergence of numerical series 661.8.2 Convergence tests 661.8.3 Examples of infinite numerical series 68

1.9 Infinite Products 711.9.1 Convergence of infinite products 711.9.2 Examples of infinite products 71

1.10 Functional Series 731.10.1 Uniform convergence 73

1.11 Power Series 741.11.1 Definition 74

Contents

1.12 Taylor Series 791.12.1 Definition and forms of remainder term 791.12.4 Order notation (Big O and little o) 80

1.13 Fourier Series 811.13.1 Definitions 81

1.14 Asymptotic Expansions 851.14.1 Introduction 851.14.2 Definition and properties of asymptotic series 86

1.15 Basic Results from the Calculus 861.15.1 Rules for differentiation 861.15.2 Integration 881.15.3 Reduction formulas 911.15.4 Improper integrals 921.15.5 Integration of rational functions 941.15.6 Elementary applications of definite integrals 96

Functions and Identities

2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 1012.1.1 Basic results 101

2.2 Logarithms and Exponentials 1122.2.1 Basic functional relationships 1122.2.2 The number e 113

2.3 The Exponential Function 1142.3.1 Series representations 114

2.4 Trigonometric Identities 1152.4.1 Trigonometric functions 115

2.5 Hyperbolic Identities 1212.5.1 Hyperbolic functions 121

2.6 The Logarithm 1262.6.1 Series representations 126

2.7 Inverse Trigonometric and Hyperbolic Functions 1282.7.1 Domains of definition and principal values 1282.7.2 Functional relations 128

2.8 Series Representations of Trigonometric and Hyperbolic Functions 1332.8.1 Trigonometric functions 1332.8.2 Hyperbolic functions 1342.8.3 Inverse trigonometric functions 1342.8.4 Inverse hyperbolic functions 135

2.9 Useful Limiting Values and Inequalities Involving ElementaryFunctions 1362.9.1 Logarithmic functions 1362.9.2 Exponential functions 1362.9.3 Trigonometric and hyperbolic functions 137

Contents

Q Derivatives of Elementary Functions

3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 1393.2 Derivatives of Trigonometric Functions 1403.3 Derivatives of Inverse Trigonometric Functions 1403.4 Derivatives of Hyperbolic Functions 1413.5 Derivatives of Inverse Hyperbolic Functions 142

A Indefinite Integrals of Algebraic Functions

4.1 Algebraic and Transcendental Functions 1454.1.1 Definitions 145

4.2 Indefinite Integrals of Rational Functions 1464.2.1 Integrands involving xn 1464.2.2 Integrands involving a + bx 1464.2.3 Integrands involving linear factors 1494.2.4 Integrands involving a2 ± b2x2 1504.2.5 Integrands involving a + bx + ex2 1534.2.6 Integrands involving a + bx3 1554.2.7 Integrands involving a + bxA 156

4.3 Nonrational Algebraic Functions 1584.3.1, Integrands containing a + bxk and ~J~x 1584.3.2 Integrands containing (a + bx)l/1 1604.3.3 Integrands containing (a + ex2)1/2 1614.3.4 Integrands containing (a + bx + cx2)l/2 164

C Indefinite Integrals of Exponential Functions

5.1 Basic Results 1675.1.1 Indefinite integrals involving eax 1675.1.2 Integrands involving the exponential functions combined with rational

functions of x 1685.1.3 Integrands involving the exponential functions combined with

trigonometric functions 169

g Indefinite Integrals of Logarithmic Functions

6.1 Combinations of Logarithms and Polynomials 1736.1.1 The logarithm 1736.1.2 Integrands involving combinations of In (a*) and powers of x 1746.1.3 Integrands involving (a + bx)m In" x 175

Contents

6.1.4 Integrands involving ln(;t2 ± a2) 1116.1.5 Integrands involving xm\n[x + (x2 ± a2)1'2] 178

J Indefinite Integrals of Hyperbolic Functions

7.1 Basic Results 1797.1.1 Integrands involving sinh(a + bx) and cosh(a + bx) 179

7.2 Integrands Involving Powers of sinh(fox) or cosh{bx) 1807.2.1 Integrands involving powers of sinh(fojc) 1807.2.2 Integrands involving powers of cosh(fe) 180

7.3 Integrands Involving (a ± bx)m sinh(cx:) or (a + bx)m cosh(cx) 1817.3.1 General results 181

7.4 Integrands Involving xm sinh"* or xm cosh"x 1837.4.1 Integrands involving xm sinh"x 1837.4.2 Integrals involving xm cosh"x 183

7.5 Integrands Involving xm sinh~"x or xm cosh""x 1837.5.1 Integrands involving xm sinh""x 1837.5.2 Integrands involving xm cosh""* 184

7.6 Integrands Involving (1 ± coshx)~m 1857.6.1 Integrands involving (1 ± coshx)~l 1857.6.2 Integrands involving (1 ± coshx)~2 185

7.7 Integrands Involving sinh(a;c)cosh~";t or cosh(a;c)sinh~"x 1857.7.1 Integrands involving sinh(a^) cosh~"x 1857.7.2 Integrands involving cosh(ax) sinh"""^ 186

7.8 Integrands Involving sinh(ajc + b) and cosh(cx 4- d) 1867.8.1 General case 1867.8.2 Special case a = c 1877.8.3 Integrands involving sinh^cosh9* 187

7.9 Integrands Involving tanh kx and coth kx 188- 7.9.1 Integrands involving tanh kx 188

7.9.2 Integrands involving coth kx 1887.10 Integrands Involving (a + bx)m sinhkx or (a + bx)m coshjb: 189

7.10.1 Integrands involving (a + bx)m sirthkx 1897.10.2 Integrands involving (a + bx)m cosh kx 189

Q Indefinite Integrals Involving Inverse Hyperbolic Function;

8.1 Basic Results 191 .8.1.1 Integrands involving products of x" and arcsinh(x/a) or

arccosh(jc/a) 191 \8.2 Integrands Involving x~" arcsinh(jc/a) or x~n arccosh(jc/a) 193

8.2.1 Integrands involving *~" arcsinh(;t/a) 1938.2.2 Integrands involving x~n arccosh(x/a) 193

Contents

8.3 Integrands Involving xn arctanh(*/a) or x" arccoth(*/a) 1948.3.1 Integrands involving x" arctanh(*/a) 1948.3.2 Integrands involving x" arccoth(;c/a) 194

8.4 Integrands Involving x~" arctanh(*/a) or x~" arccoth(*/a) 1958.4.1 Integrands involving x~n arctanh(*/a) 1958.4.2 Integrands involving *~"arccoth(*/a) 195

Q Indefinite Integrals of Trigonometric Functions

9.1 Basic Results 1979.1.1 Simplification by means of substitutions 197

9.2 Integrands Involving Powers of x and Powers of sin x or cos x 1979.2.1 Integrands involving x" sinm* 1999.2.2 Integrands involving x ""sin"1* 2009.2.3 Integrands involving xn sin""1* 2019.2.4 Integrands involving x" cos"1* 2019.2.5 Integrands involving *~" cosm* 2039.2.6 Integrands involving x" cos~m* 2039.2.7 Integrands involving xn sin*/(a + bcos*)"1 or

xncosx/(a+bsinx)m 2049.3 Integrands Involving tan x and/or cot x 205

9.3.1 Integrands involving tan" x or tan" */(tan* ± 1) 2059.3.2 Integrands involving cot" x or tan* and cot* 206

9.4 Integrands Involving sin* and cos* 2079.4.1 Integrands involving sin"1* cos"* 2079.4.2 Integrands involving sin""* 2079.4.3 Integrands involving cos""* 2089.4.4 Integrands involving sinm*/cos"* or cosm*/sin"* 2089.4.5 Integrands involving sin~m* cos~"* 210

9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powersof* 2119.5.1 Integrands involving products of (ax + b)n, sin(c* + d), and/or

cos(px+q) 2119.5.2 Integrands involving *" sin"1* or *" cosm* 211

in Indefinite Integrals of Inverse Trigonometric Functions

10.1 Integrands Involving Powers of* and Powers of Inverse TrigonometricFunctions 215 . -x

10.1.1 Integrands involving*" arcsinm(*/a) 21510.1.2 Integrands involving *~" arcsin(*/a) 21610.1.3 Integrands involving *"arccosm(*/a) 21610.1.4 Integrands involving *"" arccos(*/a) 217

Contents

10.1.5 Integrands involving *"arctan(*/a) 21710.1.6 Integrands involving *~"arctan(*/a) 21810.1.7 Integrands involving *" arccot(*/a) 21810.1.8 Integrands involving *~" arccot(*/a) 21910.1.9 Integrands involving products of rational functions and

arccot(*/a) 219

The Gamma, Beta, Pi, and Psi Functions

11.1 The Euler Integral and Limit and Infinite Product RepresentationsforT(*) 22111.1.1 Definitions and notation 22111.1.2 Special properties of T(x) 22211.1.3 Asymptotic representations of F (*) and n! 22311.1.4 Special values of T(*) 22311.1.5 The gamma function in the complex plane 22311.1.6 The psi (digamma) function 22411.1.7 The beta function 22411.1.8 Graph of T(*) and tabular values of T(*) and In T(*) 225

•J2 Elliptic Integrals and Functions

12.1 Elliptic Integrals 22912.1.1 Legendre normal forms 22912.1.2 Tabulations and trigonometric series representations of complete

elliptic integrals 23112.1.3 Tabulations and trigonometric series for E(cp, k) and F(y, k) 233

12.2 Jacobian Elliptic Functions 235--12.2.1 The functions sn u, en u, and dn u 235

12.2.2 Basic results 23512.3 Derivatives and Integrals 237

12.3.1 Derivatives of sn M, en u, and dn u 23712.3.2 Integrals involving sn u, en u, and dn u 237

12.4 Inverse Jacobian Elliptic Functions 23712.4.1 Definitions 237

Probability Integrals and the Error Function

13.1 Normal Distribution 239.13.1.1 Definitions 23913.1.2 Power series representations (*> 0) 24013.1.3 Asymptotic expansions (* ^> 0) 241

Xij Contents

13.2 The Error Function 24213.2.1 Definitions 24213.2.2 Power series representation 24213.2.3 Asymptotic expansion (* » 0) 24313.2.4 Connection between />(*) and erf* 24313.2.5 Integrals expressible in terms of erf * 24313.2.6 Derivatives of erf* 24313.2.7 Integrals of erfc * 24313.2.8 Integral and power series representation of i" erfc * 24413.2.9 Value of i" erfc * at zero 244

Fresnel Integrals, Sine and Cosine Integrals

14.1 Definitions, Series Representations, and Values at Infinity 24514.1.1 The Fresnel integrals 24514.1.2 Series representations 24514.1.3 Limiting values as * ->• oo 247

14.2 Definitions, Series Representations, and Values at Infinity 24714.2.1 Sine and cosine integrals 24714.2.2 Series representations 24714.2.3 Limiting values as * -> oo 248

•JK Definite Integrals

15.1 Integrands Involving Powers of * 24915.2 Integrands Involving Trigonometric Functions 25115.3 Integrands Involving the Exponential Function 25415.4 Integrands Involving the Hyperbolic Function 25615:5- Integrands Involving the Logarithmic Function 256

Different Forms of Fourier Series

16.1 Fourier Series for / (* ) on — n < x < n 25716.1.1 The Fourier series 257

16.2 Fourier Series for / ( * ) on -L < * < L 25816.2.1 The Fourier series 258

16.3 Fourier Series for / (* ) on a < * < b 25816.3.1 The Fourier series 258

16.4 Half-Range Fourier Cosine Series for / (* ) on 0 < * < n 25916.4.1 The Fourier series 259

16.5 Half-Range Fourier Cosine Series for / (* ) on 0 < * < L 25916.5.1 The Fourier series 259

Contents

17

16.6 Half-Range Fourier Sine Series for / (* ) on 0 < * < n 26016.6.1 The Fourier series 260

16.7 Half-Range Fourier Sine Series for / (* ) on 0 < * < L 26016.7.1 The Fourier series 260

16.8 Complex (Exponential) Fourier Series for / (* ) on — n < x < n 26016.8.1 The Fourier series 260

16.9 Complex (Exponential) Fourier Series for / (* ) on — L < x < L 26116.9.1 The Fourier series 261

16.10 Representative Examples of Fourier Series 26116.11 Fourier Series and Discontinuous Functions 265

16.11.1 Periodic extensions and convergence of Fourier series 26516.11.2 Applications to closed-form summations of numerical

series 266

Bessel Functions

17.1 Bessel's Differential Equation 26917.1.1 Different forms of Bessel's equation 269

17.2 Series Expansions for /„(*) and Yv(x) 27017.2.1 Series expansions for /„(*) and Jv(x) 27017.2.2 Series expansions for Yn(x) and Yv(x) 271

17.3 Bessel Functions of Fractional Order 27217.3.1 Bessel functions/±(n+1/2)(*) 27217.3.2 Bessel functions y±(n+i/2)(*) 272

17.4 Asymptotic Representations for Bessel Functions 27317.4.1 Asymptotic representations for large arguments 27317.4.2 Asymptotic representation for large orders 273

17.5 Zeros of Bessel Functions 27317.5.1 Zeros of 7«(*) and Fn(*) 273

17.6 Bessel's Modified Equation 27417.6.1 Different forms of Bessel's modified equation 274

17.7 Series Expansions for /„(*) and Kv{x) 27617.7.1 Series expansions for /„(*) and Iv(x) 27617.7.2 Series expansions for K0(x) and Kn(x) 276

17.8 Modified Bessel Functions of Fractional Order 27717.8.1 Modified Bessel functions/±(n+i/2)(*) 27717.8.2 Modified Bessel functions A"±(n+i/2)(*) 278

17.9 Asymptotic Representations of Modified Bessel Functions 27817.9.1 Asymptotic representations for large arguments 278

17.10 Relationships between Bessel Functions 27817.10.1 Relationships involving /„(*) and Yv(x) 27817.10.2 Relationships involving Iv(x) and Kv(x) 280

17.11 Integral Representations of /„(*), /„(*), and Kn(x) 28117.11.1 Integral representations of /„ (*) 281

Contents

17.12 Indefinite Integrals of Bessel Functions 28117.12.1 Integrals of /„(*), /„(*), and Kn(x) 281

17.13 Definite Integrals Involving Bessel Functions 28217.13.1 Definite integrals involving /„(*) and elementary

functions 28217.14 Spherical Bessel Functions 283

17.14.1 The differential equation 28317.14.2 The spherical Bessel functions jn(x) and yn(x) 28417.14.3 Recurrence relations 28417.14.4 Series representations 284

1Q Orthogonal Polynomials

18.1 Introduction 28518.1.1 Definition of a system of orthogonal polynomials 285

18.2 Legendre Polynomials />„(*) 28618.2.1 Differential equation satisfied by Pn(x) 28618.2.2 Rodrigues' formula for />„(*) 28618.2.3 Orthogonality relation for Pn(x) 28618.2.4 Explicit expressions for Pn(x) 28618.2.5 Recurrence relations satisfied by Pn(x) 28818.2.6 Generating function for Pn(x) 28918.2.7 Legendre functions of the second kind Qn(x) 289

18.3 Chebyshev Polynomials Tn(x) and £/„(*) 29018.3.1 Differential equation satisfied by Tn(x) and Un(x) 29018.3.2 Rodrigues' formulas for Tn (*) and Un (*) 29018.3.3 Orthogonality relations for Tn(x) and Un(x) 29018.3.4 Explicit expressions for Jn(*) and £/„(*) 29118.3.5 Recurrence relations satisfied by Tn (*) and Un (*) 294

-. - 18.3.6 Generating functions for Tn(x) and Un(x) 29418.4 Laguerre Polynomials Ln(x) 294

18.4.1 Differential equation satisfied by Ln{x) 29418.4.2 Rodrigues' formula for Ln(x) 29518.4.3 Orthogonality relation for Ln{x) 29518.4.4 Explicit expressions for Ln(x) 29518.4.5 Recurrence relations satisfied by Ln(x) 29518.4.6 Generating function for Ln(x) 296

18.5 Hermite Polynomials Hn(x) 29618.5.1 Differential equation satisfied by//„(*) 29618.5.2 Rodrigues' formula for Hn(x) 29618.5.3 Orthogonality relation for Hn(x) 29618.5.4 Explicit expressions for Hn(x) 29618.5.5 Recurrence relations satisfied by Hn(x) 29718.5.6 Generating function for Hn(x) 297

Contents XV

7 0 Laplace Transformation

19.1 Introduction 29919.1.1 Definition of the Laplace transform 29919.1.2 Basic properties of the Laplace transform 30019.1.3 The Dirac delta function <5(*) 30119.1.4 Laplace transform pairs 301

2Q Fourier Transforms20.1 Introduction 307

20.1.1 Fourier exponential transform 30720.1.2 Basic properties of the Fourier transforms 30820.1.3 Fourier transform pairs 30920.1.4 Fourier cosine and sine transforms 30920.1.5 Basic properties of the Fourier cosine and sine transforms 31220.1.6 Fourier cosine and sine transform pairs 312

2 7 Numerical Integration21.1 Classical Methods 315

21.1.1 Open- and closed-type formulas 31521.1.2 Composite midpoint rule (open type) 31621.1.3 Composite trapezoidal rule (closed type) 31621.1.4 Composite Simpson's rule (closed type) 31621.1.5 Newton-Cotes formulas 31721.1.6 Gaussian quadrature (open-type) 31821.1.7 Romberg integration (closed-type) 318

22 Solutions of Standard Ordinary Differential Equations22.1 Introduction 321

22.1.1 Basic definitions 32122.1.2 Linear dependence and independence 322

22.2 Separation of Variables 32322.3 Linear First-Order Equations 32322.4 Bernoulli's Equation 32422.5 Exact Equations 32522.6 Homogeneous Equations 32522.7 Linear Differential Equations 32622.8 Constant Coefficient Linear Differential Equations—Homogeneous

Case 327

Contents

22.9 Linear Homogeneous Second-Order Equation 33022.10 Constant Coefficient Linear Differential Equations—Inhomogeneous

Case 33122.11 Linear Inhomogeneous Second-Order Equation 33322.12 Determination of Particular Integrals by the Method of Undetermined

Coefficients 33422.13 The Cauchy-Euler Equation 33622.14 Legendre's Equation 33722.15 Bessel's Equations 33722.16 Power Series and Frobenius Methods 33922.17 The Hypergeometric Equation 34422.18 Numerical Methods 345

Vector Analysis23.1 Scalars and Vectors 353

23.1.1 Basic definitions 35323.1.2 Vector addition and subtraction 35523.1.3 Scaling vectors 35623.1.4 Vectors in component form 357

23.2 Scalar Products 35823.3 Vector Products 35923.4 Triple Products 36023.5 Products of Four Vectors 36123.6 Derivatives of Vector Functions of a Scalar t 36123.7 Derivatives of Vector Functions of Several Scalar Variables 36223.8 Integrals of Vector Functions of a Scalar Variable t 36323.9 Line Integrals 36423.10 Vector Integral Theorems 36623.11 A Vector Rate of Change Theorem 36823.12 Useful Vector Identities and Results 368

2& Systems of Orthogonal Coordinates24.1 Curvilinear Coordinates 369

24.1.1 Basic definitions 36924.2 Vector Operators in Orthogonal Coordinates 37124.3 Systems of Orthogonal Coordinates 371

Contents XVii

Partial Differential Equations and Special Functions

25.1 Fundamental Ideas 38125.1.1 -Classification of equations 381

25.2 Method of Separation of Variables 38525.2.1 Application to a hyperbolic problem 385

25.3 The Sturm-Liouville Problem and Special Functions 38725.4 A First-Order System and the Wave Equation 39025.5 Conservation Equations (Laws) 39125.6 The Method of Characteristics 39225.7 Discontinuous Solutions (Shocks) 39625.8 Similarity Solutions 39825.9 Burgers's Equation, the KdV Equation, and the KdVB Equation 400

2Q The z-Transform

26.1 The z-Transform and Transform Pairs 403

2J Numerical Approximation27.1 Introduction 409

27.1.1 Linear interpolation 41027.1.2 Lagrange polynomial interpolation 41027.1.3 Spline interpolation 410

27.2 Economization of Series 41127.3 Pade Approximation 413

27.4 Finite Difference Approximations to Ordinary and Partial Derivatives 415

Short Classified Reference List 419

Index 423