Advanced Calculus Explored: With Applications in Physics, Chemistry, and Beyond

Book Preface

To numerically confirm the many evaluations and results throughout this book, various integration and summation commands available in software produced by Wolfram Research, Inc. were utilized. Moreover, virtually all illustrations and graphs were produced by software produced by Wolfram Research, unless specified otherwise.
Specifically, Wolfram Desktop Version 12.0.0.0 running on a Windows 10 PC. As of the time of the release of the book, this is the latest release of Wolfram Desktop.
The commands in this book are standard and are likely to continue to work for subsequent versions. Wolfram
Research does not warrant the accuracy of the results in this book. This book’s use of Wolfram Research software does not constitute an endorsement or sponsorship by Wolfram Research, Inc. of a particular pedagogical approach or particular use of the Wolfram Research software.

Contents
About the Author 15
Preface 23
I Introductory Chapters 27
1 Differential Calculus 29
1.1 The Limit . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.1.1 L’Hopital’s Rule . . . . . . . . . . . . . . . . . 36
1.1.2 More Advanced Limits . . . . . . . . . . . . . . 40
1.2 The Derivative . . . . . . . . . . . . . . . . . . . . . . 45
1.2.1 Product Rule . . . . . . . . . . . . . . . . . . . 46
1.2.2 Quotient Rule . . . . . . . . . . . . . . . . . . . 48
1.2.3 Chain Rule . . . . . . . . . . . . . . . . . . . . 49
1.3 Exercise Problems . . . . . . . . . . . . . . . . . . . . 56
7
8 CONTENTS
2 Basic Integration 59
2.1 Riemann Integral . . . . . . . . . . . . . . . . . . . . . 60
2.2 Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . 63
2.3 The u-substitution . . . . . . . . . . . . . . . . . . . . 65
2.4 Other Problems . . . . . . . . . . . . . . . . . . . . . . 82
2.5 Exercise Problems . . . . . . . . . . . . . . . . . . . . 99
3 Feynman’s Trick 101
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2 Direct Approach . . . . . . . . . . . . . . . . . . . . . 103
3.3 Indirect Approach . . . . . . . . . . . . . . . . . . . . 128
3.4 Exercise Problems . . . . . . . . . . . . . . . . . . . . 131
4 Sums of Simple Series 135
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 136
4.2 Arithmetic and Geometric Series . . . . . . . . . . . . 136
4.3 Arithmetic-Geometric Series . . . . . . . . . . . . . . . 141
4.4 Summation by Parts . . . . . . . . . . . . . . . . . . . 146
4.5 Telescoping Series . . . . . . . . . . . . . . . . . . . . . 152
4.6 Trigonometric Series . . . . . . . . . . . . . . . . . . . 159
4.7 Exercise Problems . . . . . . . . . . . . . . . . . . . . 163
CONTENTS 9
II Series and Calculus 165
5 Prerequisites 167
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 168
5.2 Ways to Prove Convergence . . . . . . . . . . . . . . . 172
5.2.1 The Comparison Test . . . . . . . . . . . . . . 172
5.2.2 The Ratio Test . . . . . . . . . . . . . . . . . . 173
5.2.3 The Integral Test . . . . . . . . . . . . . . . . . 176
5.2.4 The Root Test . . . . . . . . . . . . . . . . . . 181
5.2.5 Dirichlet’s Test . . . . . . . . . . . . . . . . . . 184
5.3 Interchanging Summation and Integration . . . . . . . 185
6 Evaluating Series 191
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 192
6.2 Some Problems . . . . . . . . . . . . . . . . . . . . . . 193
6.2.1 Harmonic Numbers . . . . . . . . . . . . . . . . 204
6.3 Exercise Problems . . . . . . . . . . . . . . . . . . . . 214
7 Series and Integrals 215
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2 Some Problems . . . . . . . . . . . . . . . . . . . . . . 216
7.3 Exercise Problems . . . . . . . . . . . . . . . . . . . . 230
10 CONTENTS
8 Fractional Part Integrals 233
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 234
8.2 Some Problems . . . . . . . . . . . . . . . . . . . . . . 235
8.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . 255
8.4 Exercise Problems . . . . . . . . . . . . . . . . . . . . 255
III A Study in the Special Functions 257
9 Gamma Function 261
9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 262
9.2 Special Values . . . . . . . . . . . . . . . . . . . . . . . 262
9.3 Properties and Representations . . . . . . . . . . . . . 264
9.4 Some Problems . . . . . . . . . . . . . . . . . . . . . . 271
9.5 Exercise Problems . . . . . . . . . . . . . . . . . . . . 275
10 Polygamma Functions 277
10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.2 Special Values . . . . . . . . . . . . . . . . . . . . . . . 279
10.3 Properties and Representations . . . . . . . . . . . . . 280
10.4 Some Problems . . . . . . . . . . . . . . . . . . . . . . 282
10.5 Exercise Problems . . . . . . . . . . . . . . . . . . . . 294
CONTENTS 11
11 Beta Function 295
11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 296
11.2 Special Values . . . . . . . . . . . . . . . . . . . . . . . 297
11.3 Properties and Representations . . . . . . . . . . . . . 297
11.4 Some Problems . . . . . . . . . . . . . . . . . . . . . . 302
11.5 Exercise Problems . . . . . . . . . . . . . . . . . . . . 309
12 Zeta Function 311
12.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 312
12.2 Special Values . . . . . . . . . . . . . . . . . . . . . . . 312
12.3 Properties and Representations . . . . . . . . . . . . . 317
12.4 Some Problems . . . . . . . . . . . . . . . . . . . . . . 326
12.5 Exercise Problems . . . . . . . . . . . . . . . . . . . . 335
IV Applications in the Mathematical Sciences and Beyond 339
13 The Big Picture 341
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 342
13.2 Goal of the Part . . . . . . . . . . . . . . . . . . . . . 343
14 Classical Mechanics 345
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 346
12 CONTENTS
14.1.1 The Lagrange Equations . . . . . . . . . . . . . 346
14.2 The Falling Chain . . . . . . . . . . . . . . . . . . . . 347
14.3 The Pendulum . . . . . . . . . . . . . . . . . . . . . . 353
14.4 Point Mass in a Force Field . . . . . . . . . . . . . . . 357
15 Physical Chemistry 363
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 364
15.2 Sodium Chloride’s Madelung Constant . . . . . . . . . 370
15.3 The Riemann Series Theorem in Action . . . . . . . . 371
15.4 Pharmaceutical Connections . . . . . . . . . . . . . . . 377
15.5 The Debye Model . . . . . . . . . . . . . . . . . . . . . 378
16 Statistical Mechanics 381
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 382
16.2 Equations of State . . . . . . . . . . . . . . . . . . . . 383
16.3 Virial Expansion . . . . . . . . . . . . . . . . . . . . . 385
16.3.1 Lennard-Jones Potential . . . . . . . . . . . . . 386
16.4 Blackbody Radiation . . . . . . . . . . . . . . . . . . . 388
16.5 Fermi-Dirac (F-D) Statistics . . . . . . . . . . . . . . . 393
17 Miscellaneous 401
17.1 Volume of a Hypersphere of Dimension N . . . . . . . 402
CONTENTS 13
17.1.1 Spherical Coordinates . . . . . . . . . . . . . . 402
17.1.2 Calculation . . . . . . . . . . . . . . . . . . . . 404
17.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . 407
17.1.4 Applications . . . . . . . . . . . . . . . . . . . . 409
17.1.5 Mathematical Connections . . . . . . . . . . . . 410
V Appendices 413
Appendix A 415
Appendix B 421
Acknowledgements 425
Answers 427
Integral Table 435
Trigonometric Identities 439
Alphabetical Index 443