Using integration techniques is an area of undergraduate mathematics that has been shunted along the neglected road of late. With the advent of other areas of mathematics and the increased use of symbolic mathematics packages, it is possible to teach concepts that could not be offered a few years ago. However, a great deal can be learned from the underlying principles of integration, and this book can be used to teach them.
The authors state that their target level is the advanced undergraduate student and they achieve that goal. Most of the concepts can be understood by anyone who has successfully completed the three-course sequence in basic calculus. Some of the topics are integrals involving binomial coefficients, partial fractions, trigonometric functions, quartic integrals, Euler's constant, the gamma and beta functions and the Riemann zeta function.
The authors use Mathematica commands to perform many of the operations and those commands are easy to understand. Furthermore, the use of Mathematica is as a supplement, it would not be necessary to understand a single Mathematica command in order to be able to follow all of the demonstrations.
I rarely teach math classes any more, and those that I do teach are at the lower levels. However, if I was given the opportunity to teach a special topics class in mathematics, I would seriously consider offering one in the fundamental techniques of integration and use this as the text.
Author(s): George Boros, Victor Moll
Publisher: Cambridge University Press
Year: 2004
Language: English
Pages: 322
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1. Introduction......Page 17
1.2. Prime Numbers and the Factorization of n!......Page 19
1.3. The Role of Symbolic Languages......Page 23
1.4. The Binomial Theorem......Page 26
1.5. The Ascending Factorial Symbol......Page 32
1.6. The Integration of Polynomials......Page 35
2.1. Introduction......Page 41
2.2. An Elementary Example......Page 46
2.2.2. Positive Discriminant......Page 47
2.3. Wallis’ Formula......Page 48
2.4. The Solution of Polynomial Equations......Page 52
2.4.2. Cubics......Page 54
2.4.3. Quartics......Page 58
2.5. The Integration of a Biquadratic......Page 60
3.1. Introduction......Page 64
3.3. An Empirical Derivation......Page 65
3.4. Scaling and a Recursion......Page 67
3.5. A Symbolic Evaluation......Page 69
3.6. A Search in Gradshteyn and Ryzhik......Page 71
3.7. Some Consequences of the Evaluation......Page 72
3.8. A Complicated Integral......Page 74
3.8.1. Warning......Page 75
4.1. Introduction......Page 77
4.2. Taylor Series......Page 81
4.3. Taylor Series of Rational Functions......Page 83
5.1. Introduction......Page 89
5.2. The Logarithm......Page 90
5.3. Some Logarithmic Integrals......Page 97
5.4. The Number e......Page 100
5.5. Arithmetical Properties of e......Page 105
5.6. The Exponential Function......Page 107
5.7. Stirling’s Formula......Page 108
5.8. Some Definite Integrals......Page 113
5.9. Bernoulli Numbers......Page 115
5.10. Combinations of Exponentials and Polynomials......Page 119
6.1. Introduction......Page 121
6.2. The Basic Trigonometric Functions and the Existence of Pi......Page 122
6.3. Solution of Cubics and Quartics by Trigonometry......Page 127
6.4. Quadratic Denominators and Wallis’ Formula......Page 128
6.5. Arithmetical Properties of Pi......Page 133
6.6. Some Expansions in Taylor Series......Page 134
6.7. A Sequence of Polynomials Approximating tan Chi......Page 140
6.8. The Infinite Product for sin Chi......Page 142
6.9. The Cotangent and the Riemann Zeta Function......Page 145
6.10. The Case of a General Quadratic Denominator......Page 149
6.11. Combinations of Trigonometric Functions and Polynomials......Page 151
7.1. Introduction......Page 153
7.2. Reduction to a Polynomial......Page 155
7.3. A Triple Sum for the Coefficients......Page 159
7.4. The Quartic Denominators: A Crude Bound......Page 160
7.5. Closed-Form Expressions for d (m).......Page 161
7.6. A Recursion......Page 163
7.7. The Taylor Expansion of the Double Square Root......Page 166
7.8. Ramanujan’s Master Theorem and a New Class of Integrals......Page 167
7.9. A Simplified Expression for P (a)......Page 169
7.9.1. Unimodality......Page 172
7.10. The Elementary Evaluation of N (a;m)......Page 175
7.11. The Expansion of the Triple Square Root......Page 176
8.1.1. Strong Liouville Theorem (Special Case, 1835)......Page 178
8.2.2. Small Variation......Page 180
8.2.3. A Proof Using Wallis’ Formula......Page 181
8.2.5. A Proof of Kortram and Sums of Two Squares......Page 182
8.4. An Integral of Laplace......Page 187
9.1. Existence of Euler’s Constant......Page 189
9.2. A Second Proof of the Existence of Euler’s Constant......Page 190
9.3. Integral Forms for Euler’s Constant......Page 192
9.4. The Rate of Convergence to Euler’s Constant......Page 196
9.4.1. A Quicker Convergence to Euler’s Constant......Page 197
9.5. Series Representations for Euler’s Constant......Page 199
9.6. The Irrationality of Gamma......Page 200
10.1. Introduction......Page 202
10.2. The Beta Function......Page 208
10.3. Integral Representations for Gamma and Beta......Page 209
10.4. Legendre’s Duplication Formula......Page 211
10.5. An Example of Degree 4......Page 214
10.6. The Expansion of the Loggamma Function......Page 217
10.7. The Product Representation for Gamma (Chi)......Page 220
10.8. Formulas from Gradshteyn and Rhyzik (G & R)......Page 222
10.9. An Expression for the Coefficients d (m)......Page 223
10.10. Holder’s Theorem for the Gamma Function......Page 226
10.11. The Psi Function......Page 228
10.12. Integral Representations for Psi (Chi)......Page 231
10.13. Some Explicit Evaluations......Page 233
11.1. Introduction......Page 235
11.2. An Integral Representation......Page 238
11.3.2. Apostol’s Proof......Page 241
11.3.3. Calabi’s Proof......Page 242
11.3.4. Matsuoka’s Proof......Page 243
11.3.5. Boo Rim Choe’s Proof......Page 246
11.4. Apery’s Constant: Stigma(3)......Page 247
11.4.1. A Formula of Ewell......Page 248
11.4.2. A Formula of Yue and Williams......Page 249
11.5. Apery Type Formulae......Page 251
12 Logarithmic Integrals......Page 253
12.1. Polynomial Examples......Page 254
12.2. Linear Denominators......Page 255
12.3. Some Quadratic Denominators......Page 257
12.4. Products of Logarithms......Page 260
12.5. The Logsine Function......Page 261
13.1. Introduction......Page 266
13.2. Schlomilch Transformation......Page 267
13.3. Derivation of the Master Formula......Page 268
13.4. Applications of the Master Formula......Page 269
13.5. Differentiation Results......Page 273
13.6. The case a = 1......Page 274
13.7. A New Series of Examples......Page 279
13.8. New Integrals by Integration......Page 282
13.9. New Integrals by Differentiation......Page 284
A.1. Introduction......Page 287
A.2. An Introduction to WZ Methods......Page 288
A.3. A Proof of Wallis’ Formula......Page 289
Bibliography......Page 292
Index......Page 315
