Handbook of Mathematical Formulas and Integrals, Fourth Edition:Handbook of Mathematical Formulas and Integrals, Fourth Edition
Author(s): Alan Jeffrey, Hui Hui Dai
Publisher: Academic Press; 4 edition
Date : 2008
Pages : 1
Format : PDF
OCR : Yes
Quality :
Language : English
ISBN-10 : 0123742889
ISBN-13 :
The extensive additions, and the inclusion of a new chapter, has made this classic work by Jeffrey, now joined by co-author Dr. H.H. Dai, an even more essential reference for researchers and students in applied mathematics, engineering, and physics. It provides quick access to important formulas, relationships between functions, and mathematical techniques that range from matrix theory and integrals of commonly occurring functions to vector calculus, ordinary and partial differential equations, special functions, Fourier series, orthogonal polynomials, and Laplace and Fourier transforms. During the preparation of this edition full advantage was taken of the recently updated seventh edition of Gradshteyn and Ryzhiks Table of Integrals, Series, and Products and other important reference works. Suggestions from users of the third edition of the Handbook have resulted in the expansion of many sections, and because of the relevance to boundary value problems for the Laplace equation in the plane, a new chapter on conformal mapping, has been added, complete with an atlas of useful mappings.
- Comprehensive coverage in reference form of the branches of mathematics used in science and engineering
- Organized to make results involving integrals and functions easy to locate
- Results illustrated by worked examples
Review
This book would find a place on the bookshelf of a professional as a nice reference source. The level is appropriate for our physics majors for use as a reference book.
--Professor Bryan H. Suits, Physics Department, Michigan Technological University
...the 4th edition will be more useful for the students, faculty and professionals in mathematics, science and engineering. Jeffrey has written many books and handbooks and has a tremendous reputation as an author...
--Lokenath Debnath, Chair and Professor at University of Texas, Pan American
Contents
Preface xix
Preface to the Fourth Edition xxi
Notes for Handbook Users xxiii
Index of Special Functions and Notations xliii
0 Quick Reference List of Frequently Used Data 1
0.1. Useful Identities 1
0.1.1. Trigonometric Identities 1
0.1.2. Hyperbolic Identities 2
0.2. Complex Relationships 2
0.3. Constants, Binomial Coefficients and the Pochhammer Symbol 3
0.4. Derivatives of Elementary Functions 3
0.5. Rules of Differentiation and Integration 4
0.6. Standard Integrals 4
0.7. Standard Series 10
0.8. Geometry 12
1 Numerical, Algebraic, and Analytical Results for Series and Calculus 27
1.1. Algebraic Results Involving Real and Complex Numbers 27
1.1.1. Complex Numbers 27
1.1.2. Algebraic Inequalities Involving Real and Complex Numbers 28
1.2. Finite Sums 32
1.2.1. The Binomial Theorem for Positive Integral Exponents 32
1.2.2. Arithmetic, Geometric, and Arithmetic–Geometric Series 36
1.2.3. Sums of Powers of Integers 36
1.2.4. Proof by Mathematical Induction 38
1.3. Bernoulli and Euler Numbers and Polynomials 40
1.3.1. Bernoulli and Euler Numbers 40
1.3.2. Bernoulli and Euler Polynomials 46
1.3.3. The Euler–Maclaurin Summation Formula 48
1.3.4. Accelerating the Convergence of Alternating Series 49
1.4. Determinants 50
1.4.1. Expansion of Second- and Third-Order Determinants 50
1.4.2. Minors, Cofactors, and the Laplace Expansion 51
1.4.3. Basic Properties of Determinants 53
v
vi Contents
1.4.4. Jacobi’s Theorem 53
1.4.5. Hadamard’s Theorem 54
1.4.6. Hadamard’s Inequality 54
1.4.7. Cramer’s Rule 55
1.4.8. Some Special Determinants 55
1.4.9. Routh–Hurwitz Theorem 57
1.5. Matrices 58
1.5.1. Special Matrices 58
1.5.2. Quadratic Forms 62
1.5.3. Differentiation and Integration of Matrices 64
1.5.4. The Matrix Exponential 65
1.5.5. The Gerschgorin Circle Theorem 67
1.6. Permutations and Combinations 67
1.6.1. Permutations 67
1.6.2. Combinations 68
1.7. Partial Fraction Decomposition 68
1.7.1. Rational Functions 68
1.7.2. Method of Undetermined Coefficients 69
1.8. Convergence of Series 72
1.8.1. Types of Convergence of Numerical Series 72
1.8.2. Convergence Tests 72
1.8.3. Examples of Infinite Numerical Series 74
1.9. Infinite Products 77
1.9.1. Convergence of Infinite Products 77
1.9.2. Examples of Infinite Products 78
1.10. Functional Series 79
1.10.1. Uniform Convergence 79
1.11. Power Series 82
1.11.1. Definition 82
1.12. Taylor Series 86
1.12.1. Definition and Forms of Remainder Term 86
1.12.2. Order Notation (Big O and Little o) 88
1.13. Fourier Series 89
1.13.1. Definitions 89
1.14. Asymptotic Expansions 93
1.14.1. Introduction 93
1.14.2. Definition and Properties of Asymptotic Series 94
1.15. Basic Results from the Calculus 95
1.15.1. Rules for Differentiation 95
1.15.2. Integration 96
1.15.3. Reduction Formulas 99
1.15.4. Improper Integrals 101
1.15.5. Integration of Rational Functions 103
1.15.6. Elementary Applications of Definite Integrals 104
Contents vii
2 Functions and Identities 109
2.1. Complex Numbers and Trigonometric and Hyperbolic Functions 109
2.1.1. Basic Results 109
2.2. Logorithms and Exponentials 121
2.2.1. Basic Functional Relationships 121
2.2.2. The Number e 123
2.3. The Exponential Function 123
2.3.1. Series Representations 123
2.4. Trigonometric Identities 124
2.4.1. Trigonometric Functions 124
2.5. Hyperbolic Identities 132
2.5.1. Hyperbolic Functions 132
2.6. The Logarithm 137
2.6.1. Series Representations 137
2.7. Inverse Trigonometric and Hyperbolic Functions 139
2.7.1. Domains of Definition and Principal Values 139
2.7.2. Functional Relations 139
2.8. Series Representations of Trigonometric and Hyperbolic Functions 144
2.8.1. Trigonometric Functions 144
2.8.2. Hyperbolic Functions 145
2.8.3. Inverse Trigonometric Functions 146
2.8.4. Inverse Hyperbolic Functions 146
2.9. Useful Limiting Values and Inequalities Involving Elementary Functions 147
2.9.1. Logarithmic Functions 147
2.9.2. Exponential Functions 147
2.9.3. Trigonometric and Hyperbolic Functions 148
3 Derivatives of Elementary Functions 149
3.1. Derivatives of Algebraic, Logarithmic, and Exponential Functions 149
3.2. Derivatives of Trigonometric Functions 150
3.3. Derivatives of Inverse Trigonometric Functions 150
3.4. Derivatives of Hyperbolic Functions 151
3.5. Derivatives of Inverse Hyperbolic Functions 152
4 Indefinite Integrals of Algebraic Functions 153
4.1. Algebraic and Transcendental Functions 153
4.1.1. Definitions 153
4.2. Indefinite Integrals of Rational Functions 154
4.2.1. Integrands Involving xn 154
4.2.2. Integrands Involving a + bx 154
4.2.3. Integrands Involving Linear Factors 157
4.2.4. Integrands Involving a2 ± b2x2 158
4.2.5. Integrands Involving a + bx + cx2 162
viii Contents
4.2.6. Integrands Involving a + bx3 164
4.2.7. Integrands Involving a + bx4 165
4.3. Nonrational Algebraic Functions 166
4.3.1. Integrands Containing a + bxk and √x 166
4.3.2. Integrands Containing (a + bx)1/2 168
4.3.3. Integrands Containing (a + cx2)1/2 170
4.3.4. Integrands Containing a + bx + cx21/2 172
5 Indefinite Integrals of Exponential Functions 175
5.1. Basic Results 175
5.1.1. Indefinite Integrals Involving eax 175
5.1.2. Integrals Involving the Exponential Functions
Combined with Rational Functions of x 175
5.1.3. Integrands Involving the Exponential Functions
Combined with Trigonometric Functions 177
6 Indefinite Integrals of Logarithmic Functions 181
6.1. Combinations of Logarithms and Polynomials 181
6.1.1. The Logarithm 181
6.1.2. Integrands Involving Combinations of ln(ax)
and Powers of x 182
6.1.3. Integrands Involving (a + bx)m lnn x 183
6.1.4. Integrands Involving ln(x2 ± a2) 185
6.1.5. Integrands Involving xm lnx + x2 ± a21/2 186
7 Indefinite Integrals of Hyperbolic Functions 189
7.1. Basic Results 189
7.1.1. Integrands Involving sinh(a + bx) and cosh(a + bx) 189
7.2. Integrands Involving Powers of sinh(bx) or cosh(bx) 190
7.2.1. Integrands Involving Powers of sinh(bx) 190
7.2.2. Integrands Involving Powers of cosh(bx) 190
7.3. Integrands Involving (a + bx)m sinh(cx) or (a + bx)m cosh(cx) 191
7.3.1. General Results 191
7.4. Integrands Involving xm sinhnx or xm coshnx 193
7.4.1. Integrands Involving xm sinhnx 193
7.4.2. Integrands Involving xm coshnx 193
7.5. Integrands Involving xm sinhnx or xm coshnx 193
7.5.1. Integrands Involving xm sinhnx 193
7.5.2. Integrands Involving xm coshnx 194
7.6. Integrands Involving (1 ± cosh x)−m 195
7.6.1. Integrands Involving (1 ± cosh x)−1 195
7.6.2. Integrands Involving (1 ± cosh x)−2 195
Contents ix
7.7. Integrands Involving sinh(ax) cosh−n x or cosh(ax) sinh−n x 195
7.7.1. Integrands Involving sinh(ax) coshn x 195
7.7.2. Integrands Involving cosh(ax) sinhn x 196
7.8. Integrands Involving sinh(ax + b) and cosh(cx + d) 196
7.8.1. General Case 196
7.8.2. Special Case a = c 197
7.8.3. Integrands Involving sinhp x coshq x 197
7.9. Integrands Involving tanh kx and coth kx 198
7.9.1. Integrands Involving tanh kx 198
7.9.2. Integrands Involving coth kx 198
7.10. Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 199
7.10.1. Integrands Involving (a + bx)m sinh kx 199
7.10.2. Integrands Involving (a + bx)m cosh kx 199
8 Indefinite Integrals Involving Inverse Hyperbolic Functions 201
8.1. Basic Results 201
8.1.1. Integrands Involving Products of xn and
arcsinh(x/a) or arc(x/c) 201
8.2. Integrands Involving x−n arcsinh(x/a) or x−n arccosh(x/a) 202
8.2.1. Integrands Involving x−n arcsinh(x/a) 202
8.2.2. Integrands Involving x−n arccosh(x/a) 203
8.3. Integrands Involving xn arctanh(x/a) or xn arccoth(x/a) 204
8.3.1. Integrands Involving xn arctanh(x/a) 204
8.3.2. Integrands Involving xn arccoth(x/a) 204
8.4. Integrands Involving x−n arctanh(x/a) or x−n arccoth(x/a) 205
8.4.1. Integrands Involving x−n arctanh(x/a) 205
8.4.2. Integrands Involving x−n arccoth(x/a) 205
9 Indefinite Integrals of Trigonometric Functions 207
9.1. Basic Results 207
9.1.1. Simplification by Means of Substitutions 207
9.2. Integrands Involving Powers of x and Powers of sin x or cos x 209
9.2.1. Integrands Involving xn sinm x 209
9.2.2. Integrands Involving x−n sinm x 210
9.2.3. Integrands Involving xn sin−m x 211
9.2.4. Integrands Involving xn cosm x 212
9.2.5. Integrands Involving x−n cosm x 213
9.2.6. Integrands Involving xn cos−m x 213
9.2.7. Integrands Involving xn sin x/(a + b cos x)m
or xn cos x/(a + b sin x)m 214
9.3. Integrands Involving tan x and/or cot x 215
9.3.1. Integrands Involving tann x or tann x/(tan x ± 1) 215
9.3.2. Integrands Involving cotn x or tan x and cot x 216
x Contents
9.4. Integrands Involving sin x and cos x 217
9.4.1. Integrands Involving sinm x cosn x 217
9.4.2. Integrands Involving sin−n x 217
9.4.3. Integrands Involving cos−n x 218
9.4.4. Integrands Involving sinm x/ cosn x cosm x/ sinn x 218
9.4.5. Integrands Involving sin−m x cos−n x 220
9.5. Integrands Involving Sines and Cosines with Linear
Arguments and Powers of x 221
9.5.1. Integrands Involving Products of (ax + b)n, sin(cx + d),
and/or cos(px + q) 221
9.5.2. Integrands Involving xn sinm x or xn cosm x 222
10 Indefinite Integrals of Inverse Trigonometric Functions 225
10.1. Integrands Involving Powers of x and Powers of Inverse Trigonometric
Functions 225
10.1.1. Integrands Involving xn arcsinm(x/a) 225
10.1.2. Integrands Involving x−n arcsin(x/a) 226
10.1.3. Integrands Involving xn arccosm(x/a) 226
10.1.4. Integrands Involving x−n arccos(x/a) 227
10.1.5. Integrands Involving xn arctan(x/a) 227
10.1.6. Integrands Involving x−n arctan(x/a) 227
10.1.7. Integrands Involving xn arccot(x/a) 228
10.1.8. Integrands Involving x−n arccot(x/a) 228
10.1.9. Integrands Involving Products of Rational
Functions and arccot(x/a) 229
11 The Gamma, Beta, Pi, and Psi Functions, and the Incomplete
Gamma Functions 231
11.1. The Euler Integral Limit and Infinite Product Representations
for the Gamma Function (x). The Incomplete Gamma Functions
(α, x) and γ(α, x) 231
11.1.1. Definitions and Notation 231
11.1.2. Special Properties of (x) 232
11.1.3. Asymptotic Representations of (x) and n! 233
11.1.4. Special Values of (x) 233
11.1.5. The Gamma Function in the Complex Plane 233
11.1.6. The Psi (Digamma) Function 234
11.1.7. The Beta Function 235
11.1.8. Graph of (x) and Tabular Values of (x) and ln (x) 235
11.1.9. The Incomplete Gamma Function 236
12 Elliptic Integrals and Functions 241
12.1. Elliptic Integrals 241
12.1.1. Legendre Normal Forms 241
Contents xi
12.1.2. Tabulations and Trigonometric Series Representations
of Complete Elliptic Integrals 243
12.1.3. Tabulations and Trigonometric Series for E(ϕ, k) and F(ϕ, k) 245
12.2. Jacobian Elliptic Functions 247
12.2.1. The Functions sn u, cn u, and dn u 247
12.2.2. Basic Results 247
12.3. Derivatives and Integrals 249
12.3.1. Derivatives of sn u, cn u, and dn u 249
12.3.2. Integrals Involving sn u, cn u, and dn u 249
12.4. Inverse Jacobian Elliptic Functions 250
12.4.1. Definitions 250
13 Probability Distributions and Integrals,
and the Error Function 253
13.1. Distributions 253
13.1.1. Definitions 253
13.1.2. Power Series Representations (x ≥ 0) 256
13.1.3. Asymptotic Expansions (x 0) 256
13.2. The Error Function 257
13.2.1. Definitions 257
13.2.2. Power Series Representation 257
13.2.3. Asymptotic Expansion (x 0) 257
13.2.4. Connection Between P(x) and erf x 258
13.2.5. Integrals Expressible in Terms of erf x 258
13.2.6. Derivatives of erf x 258
13.2.7. Integrals of erfc x 258
13.2.8. Integral and Power Series Representation of in erfc x 259
13.2.9. Value of in erfc x at zero 259
14 Fresnel Integrals, Sine and Cosine Integrals 261
14.1. Definitions, Series Representations, and Values at Infinity 261
14.1.1. The Fresnel Integrals 261
14.1.2. Series Representations 261
14.1.3. Limiting Values as x→∞ 263
14.2. Definitions, Series Representations, and Values at Infinity 263
14.2.1. Sine and Cosine Integrals 263
14.2.2. Series Representations 263
14.2.3. Limiting Values as x→∞ 264
15 Definite Integrals 265
15.1. Integrands Involving Powers of x 265
15.2. Integrands Involving Trigonometric Functions 267
15.3. Integrands Involving the Exponential Function 270
15.4. Integrands Involving the Hyperbolic Function 273
xii Contents
15.5. Integrands Involving the Logarithmic Function 273
15.6. Integrands Involving the Exponential Integral Ei(x) 274
16 Different Forms of Fourier Series 275
16.1. Fourier Series for f(x) on −π ≤ x ≤ π 275
16.1.1. The Fourier Series 275
16.2. Fourier Series for f(x) on −L ≤ x ≤ L 276
16.2.1. The Fourier Series 276
16.3. Fourier Series for f(x) on a ≤ x ≤ b 276
16.3.1. The Fourier Series 276
16.4. Half-Range Fourier Cosine Series for f(x) on 0 ≤ x ≤ π 277
16.4.1. The Fourier Series 277
16.5. Half-Range Fourier Cosine Series for f(x) on 0 ≤ x ≤ L 277
16.5.1. The Fourier Series 277
16.6. Half-Range Fourier Sine Series for f(x) on 0 ≤ x ≤ π 278
16.6.1. The Fourier Series 278
16.7. Half-Range Fourier Sine Series for f(x) on 0 ≤ x ≤ L 278
16.7.1. The Fourier Series 278
16.8. Complex (Exponential) Fourier Series for f(x) on −π ≤ x ≤ π 279
16.8.1. The Fourier Series 279
16.9. Complex (Exponential) Fourier Series for f(x) on −L ≤ x ≤ L 279
16.9.1. The Fourier Series 279
16.10. Representative Examples of Fourier Series 280
16.11. Fourier Series and Discontinuous Functions 285
16.11.1. Periodic Extensions and Convergence of Fourier Series 285
16.11.2. Applications to Closed-Form Summations
of Numerical Series 285
17 Bessel Functions 289
17.1. Bessel’s Differential Equation 289
17.1.1. Different Forms of Bessel’s Equation 289
17.2. Series Expansions for Jν(x) and Yν(x) 290
17.2.1. Series Expansions for Jn(x) and Jν(x) 290
17.2.2. Series Expansions for Yn(x) and Yν(x) 291
17.2.3. Expansion of sin(x sin θ) and cos(x sin θ) in
Terms of Bessel Functions 292
17.3. Bessel Functions of Fractional Order 292
17.3.1. Bessel Functions J±(n+1/2)(x) 292
17.3.2. Bessel Functions Y±(n+1/2)(x) 293
17.4. Asymptotic Representations for Bessel Functions 294
17.4.1. Asymptotic Representations for Large Arguments 294
17.4.2. Asymptotic Representation for Large Orders 294
17.5. Zeros of Bessel Functions 294
17.5.1. Zeros of Jn(x) and Yn(x) 294
Contents xiii
17.6. Bessel’s Modified Equation 294
17.6.1. Different Forms of Bessel’s Modified Equation 294
17.7. Series Expansions for Iν(x) and Kν(x) 297
17.7.1. Series Expansions for In(x) and Iν(x) 297
17.7.2. Series Expansions for K0(x) and Kn(x) 298
17.8. Modified Bessel Functions of Fractional Order 298
17.8.1. Modified Bessel Functions I±(n+1/2)(x) 298
17.8.2. Modified Bessel Functions K±(n+1/2)(x) 299
17.9. Asymptotic Representations of Modified Bessel Functions 299
17.9.1. Asymptotic Representations for Large Arguments 299
17.10. Relationships Between Bessel Functions 299
17.10.1. Relationships Involving Jν(x) and Yν(x) 299
17.10.2. Relationships Involving Iν(x) and Kν(x) 301
17.11. Integral Representations of Jn(x), In(x), and Kn(x) 302
17.11.1. Integral Representations of Jn(x) 302
17.12. Indefinite Integrals of Bessel Functions 302
17.12.1. Integrals of Jn(x), In(x), and Kn(x) 302
17.13. Definite Integrals Involving Bessel Functions 303
17.13.1. Definite Integrals Involving Jn(x) and Elementary Functions 303
17.14. Spherical Bessel Functions 304
17.14.1. The Differential Equation 304
17.14.2. The Spherical Bessel Function jn(x) and yn(x) 305
17.14.3. Recurrence Relations 306
17.14.4. Series Representations 306
17.14.5. Limiting Values as x→0 306
17.14.6. Asymptotic Expansions of jn(x) and yn(x)
When the Order n Is Large 307
17.15. Fourier-Bessel Expansions 307
18 Orthogonal Polynomials 309
18.1. Introduction 309
18.1.1. Definition of a System of Orthogonal Polynomials 309
18.2. Legendre Polynomials Pn(x) 310
18.2.1. Differential Equation Satisfied by Pn(x) 310
18.2.2. Rodrigues’ Formula for Pn(x) 310
18.2.3. Orthogonality Relation for Pn(x) 310
18.2.4. Explicit Expressions for Pn(x) 310
18.2.5. Recurrence Relations Satisfied by Pn(x) 312
18.2.6. Generating Function for Pn(x) 313
18.2.7. Legendre Functions of the Second Kind Qn(x) 313
18.2.8. Definite Integrals Involving Pn(x) 315
18.2.9. Special Values 315
xiv Contents
18.2.10. Associated Legendre Functions 316
18.2.11. Spherical Harmonics 318
18.3. Chebyshev Polynomials Tn(x) and Un(x) 320
18.3.1. Differential Equation Satisfied by Tn(x) and Un(x) 320
18.3.2. Rodrigues’ Formulas for Tn(x) and Un(x) 320
18.3.3. Orthogonality Relations for Tn(x) and Un(x) 320
18.3.4. Explicit Expressions for Tn(x) and Un(x) 321
18.3.5. Recurrence Relations Satisfied by Tn(x) and Un(x) 325
18.3.6. Generating Functions for Tn(x) and Un(x) 325
18.4. Laguerre Polynomials Ln(x) 325
18.4.1. Differential Equation Satisfied by Ln(x) 325
18.4.2. Rodrigues’ Formula for Ln(x) 325
18.4.3. Orthogonality Relation for Ln(x) 326
18.4.4. Explicit Expressions for Ln(x) and xn in
Terms of Ln(x) 326
18.4.5. Recurrence Relations Satisfied by Ln(x) 327
18.4.6. Generating Function for Ln(x) 327
18.4.7. Integrals Involving Ln(x) 327
18.4.8. Generalized (Associated) Laguerre Polynomials
L(α)
n (x) 327
18.5. Hermite Polynomials Hn(x) 329
18.5.1. Differential Equation Satisfied by Hn(x) 329
18.5.2. Rodrigues’ Formula for Hn(x) 329
18.5.3. Orthogonality Relation for Hn(x) 330
18.5.4. Explicit Expressions for Hn(x) 330
18.5.5. Recurrence Relations Satisfied by Hn(x) 330
18.5.6. Generating Function for Hn(x) 331
18.5.7. Series Expansions of Hn(x) 331
18.5.8. Powers of x in Terms of Hn(x) 331
18.5.9. Definite Integrals 331
18.5.10. Asymptotic Expansion for Large n 332
18.6. Jacobi Polynomials P(α,β)
n (x) 332
18.6.1. Differential Equation Satisfied by P(α,β)
n (x) 333
18.6.2. Rodrigues’ Formula for P(α,β)
n (x) 333
18.6.3. Orthogonality Relation for P(α,β)
n (x) 333
18.6.4. A Useful Integral Involving P(α,β)
n (x) 333
18.6.5. Explicit Expressions for P(α,β)
n (x) 333
18.6.6. Differentiation Formulas for P(α,β)
n (x) 334
18.6.7. Recurrence Relation Satisfied by P(α,β)
n (x) 334
18.6.8. The Generating Function for P(α,β)
n (x) 334
18.6.9. Asymptotic Formula for P(α,β)
n (x) for Large n 335
18.6.10. Graphs of the Jacobi Polynomials P(α,β)
n (x) 335
Contents xv
19 Laplace Transformation 337
19.1. Introduction 337
19.1.1. Definition of the Laplace Transform 337
19.1.2. Basic Properties of the Laplace Transform 338
19.1.3. The Dirac Delta Function δ(x) 340
19.1.4. Laplace Transform Pairs 340
19.1.5. Solving Initial Value Problems by the Laplace
Transform 340
20 Fourier Transforms 353
20.1. Introduction 353
20.1.1. Fourier Exponential Transform 353
20.1.2. Basic Properties of the Fourier Transforms 354
20.1.3. Fourier Transform Pairs 355
20.1.4. Fourier Cosine and Sine Transforms 357
20.1.5. Basic Properties of the Fourier Cosine and Sine
Transforms 358
20.1.6. Fourier Cosine and Sine Transform Pairs 359
21 Numerical Integration 363
21.1. Classical Methods 363
21.1.1. Open- and Closed-Type Formulas 363
21.1.2. Composite Midpoint Rule (open type) 364
21.1.3. Composite Trapezoidal Rule (closed type) 364
21.1.4. Composite Simpson’s Rule (closed type) 364
21.1.5. Newton–Cotes formulas 365
21.1.6. Gaussian Quadrature (open-type) 366
21.1.7. Romberg Integration (closed-type) 367
22 Solutions of Standard Ordinary Differential
Equations 371
22.1. Introduction 371
22.1.1. Basic Definitions 371
22.1.2. Linear Dependence and Independence 371
22.2. Separation of Variables 373
22.3. Linear First-Order Equations 373
22.4. Bernoulli’s Equation 374
22.5. Exact Equations 375
22.6. Homogeneous Equations 376
22.7. Linear Differential Equations 376
22.8. Constant Coefficient Linear Differential
Equations—Homogeneous Case 377
22.9. Linear Homogeneous Second-Order Equation 381
xvi Contents
22.10. Linear Differential Equations—Inhomogeneous Case
and the Green’s Function 382
22.11. Linear Inhomogeneous Second-Order Equation 389
22.12. Determination of Particular Integrals by the Method
of Undetermined Coefficients 390
22.13. The Cauchy–Euler Equation 393
22.14. Legendre’s Equation 394
22.15. Bessel’s Equations 394
22.16. Power Series and Frobenius Methods 396
22.17. The Hypergeometric Equation 403
22.18. Numerical Methods 404
23 Vector Analysis 415
23.1. Scalars and Vectors 415
23.1.1. Basic Definitions 415
23.1.2. Vector Addition and Subtraction 417
23.1.3. Scaling Vectors 418
23.1.4. Vectors in Component Form 419
23.2. Scalar Products 420
23.3. Vector Products 421
23.4. Triple Products 422
23.5. Products of Four Vectors 423
23.6. Derivatives of Vector Functions of a Scalar t 423
23.7. Derivatives of Vector Functions of Several Scalar Variables 425
23.8. Integrals of Vector Functions of a Scalar Variable t 426
23.9. Line Integrals 427
23.10. Vector Integral Theorems 428
23.11. A Vector Rate of Change Theorem 431
23.12. Useful Vector Identities and Results 431
24 Systems of Orthogonal Coordinates 433
24.1. Curvilinear Coordinates 433
24.1.1. Basic Definitions 433
24.2. Vector Operators in Orthogonal Coordinates 435
24.3. Systems of Orthogonal Coordinates 436
25 Partial Differential Equations and Special Functions 447
25.1. Fundamental Ideas 447
25.1.1. Classification of Equations 447
25.2. Method of Separation of Variables 451
25.2.1. Application to a Hyperbolic Problem 451
25.3. The Sturm–Liouville Problem and Special Functions 456
25.4. A First-Order System and the Wave Equation 456
Contents xvii
25.5. Conservation Equations (Laws) 457
25.6. The Method of Characteristics 458
25.7. Discontinuous Solutions (Shocks) 462
25.8. Similarity Solutions 465
25.9. Burgers’s Equation, the KdV Equation, and the KdVB Equation 467
25.10. The Poisson Integral Formulas 470
25.11. The Riemann Method 471
26 Qualitative Properties of the Heat and Laplace Equation 473
26.1. The Weak Maximum/Minimum Principle for the Heat Equation 473
26.2. The Maximum/Minimum Principle for the Laplace Equation 473
26.3. Gauss Mean Value Theorem for Harmonic Functions in the Plane 473
26.4. Gauss Mean Value Theorem for Harmonic Functions in Space 474
27 Solutions of Elliptic, Parabolic, and Hyperbolic Equations 475
27.1. Elliptic Equations (The Laplace Equation) 475
27.2. Parabolic Equations (The Heat or Diffusion Equation) 482
27.3. Hyperbolic Equations (Wave Equation) 488
28 The z-Transform 493
28.1. The z-Transform and Transform Pairs 493
29 Numerical Approximation 499
29.1. Introduction 499
29.1.1. Linear Interpolation 499
29.1.2. Lagrange Polynomial Interpolation 500
29.1.3. Spline Interpolation 500
29.2. Economization of Series 501
29.3. Pad´e Approximation 503
29.4. Finite Difference Approximations to Ordinary and Partial Derivatives 505
30 Conformal Mapping and Boundary Value Problems 509
30.1. Analytic Functions and the Cauchy-Riemann Equations 509
30.2. Harmonic Conjugates and the Laplace Equation 510
30.3. Conformal Transformations and Orthogonal Trajectories 510
30.4. Boundary Value Problems 511
30.5. Some Useful Conformal Mappings 512
Short Classified Reference List 525
Index
:27bb :11bb :31bb :27bb
dq
Handbook of Mathematical Formulas and Integrals Fourth Edition
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[ 本帖最后由 drjiachen 于 2008-12-11 11:03 编辑 ]
不错不错:27bb :29bb
等了一个上午,终于把附件等到了,:29bb
這本歸納了許多常用的數學公式
如三角函數 微積分等等
絕對是本不可錯過的好書
感謝樓主大大無私分享!
wa,好新的书。。。。。。。。。。。。。
就一个附件？还以为很长都不打算下了，O(∩_∩)O哈哈~
看起来是个很棒的工具书！！ 感谢分享～
很好的手册，谢谢楼主发布
:11bb :11bb :11bb
感谢楼主
:11bb :11bb :11bb :11bb :11bb
书不错！！！谢谢分享！！！:11bb
很好的书，感觉数学是很重要的基础
有本手册翻翻确实很方便:11bb
似的发送到发送发送飞洒似的萨发送到发送似的
谢谢楼主分享！:27bb :29bb
谢谢楼主分享！:27bb :29bb
最近真是好处层出不穷呀！
哈哈哈哈哈哈哈 谢谢楼主了 :29bb
wa,好新的书。。。。。。。。。。。。
think you very much!
:29bb :29bb :29bb :29bb
看起来是个很棒的工具书！！ 感谢分享～
:27bb :27bb :27bb :27bb
:11bb :27bb :29bb :30bb :31bb
搞工科的数学工具书是不可少的
:31bb :31bb :31bb
竟然还有这样的书。。。非常感谢楼主
很想知道谁能牢记书中全部内容。。。
好书，好！好！好！感谢楼主分享!!!!!!!!!
thank you for the book
谢谢楼主分享！:30bb :30bb :30bb
thanks for so many formulars to ref!!!!
感謝牛大分享！這本有用的書～～學習一下
不错不错:31bb :30bb :11bb
:16bb :16bb :27bb :27bb :27bb :53bb :53bb :53bb
:11bb :27bb :29bb :30bb :31bb
谢谢了:11bb :11bb
数学是很重要的基础
给此楼层加分
這本歸納了許多常用的數學公式
如三角函數 微積分等等
絕對是本不可錯過的好書
感謝樓主大大無私
Thank you very much!
好书！
谢谢1
很好的手册，谢谢楼主发布
thank you for your help
手册，看看
對於常用公式推導很用幫助
好書一本
多謝
:30bb:29bb
谢谢楼主的号好东西。
是一本经典的工具书！值得收藏！
是一本很经典的工具书！ 可以收藏！
呵呵参考书很有用
基础数学必不可少!!!支持支持!
看起来是个很棒的工具书！！ 感谢分享～
谢谢楼主了
謝謝樓漲分享
thanks very much
好书，看看
大吗？有个中文的数学手册
很好的数学手册！
很好的数学手册！
很好的数学手册！
很好的数学手册！
謝謝分享!
是本好书，非常感谢！
good job
thanks
Thanks for your sharing.
不错不错！！！
内容很吸引人！
正需要查呢
好人，好书！
工具书，谢谢
发布什么都瞒不过如果
Handbook of Mathematical Formulas and Integrals, Fourth Edition: ha.jpg
