Handbook of Mathematical Formulas and Integrals
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Table of contents
- Title pageiii
- Copyright Pageiv
- Contentsv
- Prefacexix
- Preface to the Second Editionxxi
- Index of Special Functions and Notationsxxiii
- Chapter 0. Quick Reference List of Frequently Used Data1
- 0.1 Useful Identities1
- 0.2 Complex Relationships2
- 0.3 Constants2
- 0.4 Derivatives of Elementary Functions3
- 0.5 Rules of Differentiation and Integration3
- 0.6 Standard Integrals4
- 0.7 Standard Series11
- 0.8 Geometry13
- Chapter 1. Numerical, Algebraic, and Analytical Results for Series and Calculus25
- 1.1 Algebraic Results Involving Real and Complex Numbers25
- 1.2 Finite Sums29
- 1.3 Bernoulli and Euler Numbers and Polynomials37
- 1.4 Determinants47
- 1.5 Matrices55
- 1.6 Permutations and Combinations62
- 1.7 Partial Fraction Decomposition63
- 1.8 Convergence of Series66
- 1.9 Infinite Products71
- 1.10 Functional Series73
- 1.11 Power Series74
- 1.12 Taylor Series79
- 1.13 Fourier Series81
- 1.14 Asymptotic Expansions85
- 1.15 Basic Results from the Calculus86
- Chapter 2. Functions and Identities101
- 2.1 Complex Numbers and Trigonometric and Hyperbolic Functions101
- 2.2 Logarithms and Exponentials112
- 2.3 The Exponential Function114
- 2.4 Trigonometric Identities115
- 2.5 Hyperbolic Identities121
- 2.6 The Logarithm126
- 2.7 Inverse Trigonometric and Hyperbolic Functions128
- 2.8 Series Representations of Trigonometric and Hyperbolic Functions133
- 2.9 Useful Limiting Values and Inequalities Involving Elementary Functions136
- Chapter 3. Derivatives of Elementary Functions139
- 3.1 Derivatives of Algebraic, Logarithmic, and Exponential Functions139
- 3.2 Derivatives of Trigonometric Functions140
- 3.3 Derivatives of Inverse Trigonometric Functions140
- 3.4 Derivatives of Hyperbolic Functions141
- 3.5 Derivatives of Inverse Hyperbolic Functions142
- Chapter 4. Indefinite Integrals of Algebraic Functions145
- 4.1 Algebraic and Transcendental Functions145
- 4.2 Indefinite Integrals of Rational Functions146
- 4.3 Nonrational Algebraic Functions158
- Chapter 5 Indefinite Integrals of Exponential Functions167
- 5.1 Basic Results167
- Chapter 6. Indefinite Integrals of Logarithmic Functions173
- 6.1 Combinations of Logarithms and Polynomials173
- Chapter 7. Indefinite Integrals of Hyperbolic Functions179
- 7.1 Basic Results179
- 7.2 Integrands Involving Powers of sinh(bx) or cosh(bx)180
- 7.3 Integrands Involving (a ± bx)m sinh(cx) or (a + bx)m cosh(cx)181
- 7.4 Integrands Involving xm sinhnx or xm coshnx183
- 7.5 Integrands Involving xm sinh-nx or xm cosh-nx183
- 7.6 Integrands Involving (1 ± cosh x)-m185
- 7.7 Integrands Involving sinh(ax)cosh-nx or cosh(ax)sinh-nx185
- 7.8 Integrands Involving sinh(ax + b) and cosh(cx + d)186
- 7.9 Integrands Involving tanh kx and coth kx188
- 7.10 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx189
- Chapter 8. Indefinite Integrals Involving Inverse Hyperbolic Functions191
- 8.1 Basic Results191
- 8.2 Integrands Involving x-n arcsinh(x/a) or x-n arccosh(x/a)193
- 8.3 Integrands Involving xn arctanh(x/a) or xn arccoth(x/a)194
- 8.4 Integrands Involving x-n arctanh(x/a) or x-n arccoth(x/a)195
- Chapter 9. Indefinite Integrals of Trigonometric Functions197
- 9.1 Basic Results197
- 9.2 Integrands Involving Powers of x and Powers of sin x or cos x199
- 9.3 Integrands Involving tan x and/or cot x205
- 9.4 Integrands Involving sin x and cos x207
- 9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 211211
- Chapter 10. Indefinite Integrals of Inverse Trigonometric Functions215
- 10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions215
- Chapter 11. The Gamma, Beta, Pi, and Psi Functions221
- 11.1 The Euler Integral and Limit and Infinite Product Representations for (x)221
- Chapter 12. Elliptic Integrals and Functions229
- 12.1 Elliptic Integrals229
- 12.2 Jacobian Elliptic Functions235
- 12.3 Derivatives and Integrals237
- 12.4 Inverse Jacobian Elliptic Functions237
- Chapter 13. Probability Integrals and the Error Function239
- 13.1 Normal Distribution239
- 13.2 The Error Function242
- Chapter 14. Fresnel Integrals, Sine and Cosine Integrals245
- 14.1 Definitions, Series Representations, and Values at Infinity245
- 14.2 Definitions, Series Representations, and Values at Innity247
- Chapter 15. Definite Integrals249
- 15.1 Integrands Involving Powers of x249
- 15.2 Integrands Involving Trigonometric Functions251
- 15.3 Integrands Involving the Exponential Function254
- 15.4 Integrands Involving the Hyperbolic Function256
- 15.5 Integrands Involving the Logarithmic Function256
- Chapter 16. Different Forms of Fourier Series257
- 16.1 Fourier Series for f (x) on π ≤ x ≤ π257
- 16.2 Fourier Series for f (x) on L ≤ x ≤ L258
- 16.3 Fourier Series for f (x) on a ≤ x ≤ b258
- 16.4 Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ π259
- 16.5 Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ L259
- 16.6 Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ π260
- 16.7 Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ L260
- 16.8 Complex (Exponential) Fourier Series for f (x) on π ≤ x ≤ π260
- 16.9 Complex (Exponential) Fourier Series for f (x) on L ≤ x ≤ L261
- 16.10 Representative Examples of Fourier Series261
- 16.11 Fourier Series and Discontinuous Functions265
- Chapter 17. Bessel Functions269
- 17.1 Bessel’s Differential Equation269
- 17.2 Series Expansions for Jν(x) and Yν(x)270
- 17.3 Bessel Functions of Fractional Order272
- 17.4 Asymptotic Representations for Bessel Functions273
- 17.5 Zeros of Bessel Functions273
- 17.6 Bessel’s Modified Equation274
- 17.7 Series Expansions for Iν(x) and Kν(x)276
- 17.8 Modified Bessel Functions of Fractional Order277
- 17.9 Asymptotic Representations of Modified Bessel Functions278
- 17.10 Relationships between Bessel Functions278
- 17.11 Integral Representations of Jn(x), In(x), and Kn(x)281
- 17.12 Indefinite Integrals of Bessel Functions281
- 17.13 Definite Integrals Involving Bessel Functions282
- 17.14 Spherical Bessel Functions283
- Chapter 18. Orthogonal Polynomials285
- 18.1 Introduction285
- 18.2 Legendre Polynomials Pn(x)286
- 18.3 Chebyshev Polynomials Tn(x) and Un(x)290
- 18.4 Laguerre Polynomials Ln(x)294
- 18.5 Hermite Polynomials Hn(x)296
- Chapter 19. Laplace Transformation299
- 19.1 Introduction299
- Chapter 20. Fourier Transforms307
- 20.1 Introduction307
- Chapter 21. Numerical Integration315
- 21.1 Classical Methods315
- Chapter 22. Solutions of Standard Ordinary Differential Equations321
- 22.1 Introduction321
- 22.2 Separation of Variables323
- 22.3 Linear First-Order Equations323
- 22.4 Bernoulli’s Equation324
- 22.5 Exact Equations325
- 22.6 Homogeneous Equations325
- 22.7 Linear Differential Equations326
- 22.8 Constant Coefficient Linear Differential Equations„Homogeneous Case327
- 22.9 Linear Homogeneous Second-Order Equation330
- 22.10 Constant Coefficient Linear Differential Equations„Inhomogeneous Case331
- 22.11 Linear Inhomogeneous Second-Order Equation333
- 22.12 Determination of Particular Integrals by the Method of Undetermined Coefficients334
- 22.13 The Cauchy–Euler Equation336
- 22.14 Legendre’s Equation337
- 22.15 Bessel’s Equations337
- 22.16 Power Series and Frobenius Methods339
- 22.17 The Hypergeometric Equation344
- 22.18 Numerical Methods345
- Chapter 23. Vector Analysis353
- 23.1 Scalars and Vectors353
- 23.2 Scalar Products358
- 23.3 Vector Products359
- 23.4 Triple Products360
- 23.5 Products of Four Vectors361
- 23.6 Derivatives of Vector Functions of a Scalar t361
- 23.7 Derivatives of Vector Functions of Several Scalar Variables362
- 23.8 Integrals of Vector Functions of a Scalar Variable t363
- 23.9 Line Integrals364
- 23.10 Vector Integral Theorems366
- 23.11 A Vector Rate of Change Theorem368
- 23.12 Useful Vector Identities and Results368
- Chapter 24 Systems of Orthogonal Coordinates369
- 24.1 Curvilinear Coordinates369
- 24.2 Vector Operators in Orthogonal Coordinates371
- 24.3 Systems of Orthogonal Coordinates371
- Chapter 25. Partial Differential Equations and Special Functions381
- 25.1 Fundamental Ideas381
- 25.2 Method of Separation of Variables385
- 25.3 The Sturm–Liouville Problem and Special Functions387
- 25.4 A First-Order System and the Wave Equation390
- 25.5 Conservation Equations (Laws)391
- 25.6 The Method of Characteristics392
- 25.7 Discontinuous Solutions (Shocks)396
- 25.8 Similarity Solutions398
- 25.9 Burgers’s Equation, the KdV Equation, and the KdVB Equation400
- Chapter 26. The z-Transform403
- 26.1 The z -Transform and Transform Pairs403
- Chapter 27. Numerical Approximation409
- 27.1 Introduction409
- 27.2 Economization of Series411
- 27.3 Padé Approximation413
- 27.4 Finite Difference Approximations to Ordinary and Partial Derivatives415
- Short Classified Reference List419
- Index423
The updated Handbook is an essential reference for researchers and students in applied mathematics, engineering, and physics. It provides quick access to important formulas, relations, and methods from algebra, trigonometric and exponential functions, combinatorics, probability, matrix theory, calculus and vector calculus, ordinary and partial differential equations, Fourier series, orthogonal polynomials, and Laplace transforms. Many of the entries are based upon the updated sixth edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and other important reference works.
The Third Edition has new chapters covering solutions of elliptic, parabolic and hyperbolic equations and qualitative properties of the heat and Laplace equation.
Key Features:
* Comprehensive coverage of frequently used integrals, functions and fundamental mathematical results
* Contents selected and organized to suit the needs of students, scientists, and engineers
* Contains tables of Laplace and Fourier transform pairs
* New section on numerical approximation
* New section on the z-transform
* Easy reference system
The Third Edition has new chapters covering solutions of elliptic, parabolic and hyperbolic equations and qualitative properties of the heat and Laplace equation.
Key Features:
* Comprehensive coverage of frequently used integrals, functions and fundamental mathematical results
* Contents selected and organized to suit the needs of students, scientists, and engineers
* Contains tables of Laplace and Fourier transform pairs
* New section on numerical approximation
* New section on the z-transform
* Easy reference system