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Description
Hi! My name is
Hamza Alsamraee, and I am a senior (12th grade) at Centreville High School,
Virginia. I have always had an affinity for mathematics, and from a very young
age was motivated to pursue my curiosity. When I entered a new school in 7th
grade after moving, I encountered some new mathematics I was unequipped for.
Namely, I did not know what a linear equation even was! I was rather
low-spirited, as I was stuck in an ever-lasting loop of confusion in class.
My mother and
father soon began teaching me to the best of their ability. Fortunately, their
efforts were effective, and I got a B on my first linear equations test! It was
a huge improvement from being totally lost, but I wanted to know more. I did
not care much about the grade, but I did care that I did not completely master the
material.
I soon entered in
a period of rapid learning, delving into curriculums significantly beyond my
coursework simply for the sake of mastering higher mathematics. It seemed to me
that the more I explored the field, the more beautiful the results were.
Table of 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
78 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 . . . 163CONTENTS 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 . . . 23010 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 . . . 294CONTENTS 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
. . . 34612 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 . . .
402CONTENTS 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
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