Numbers and functions: from a classical-experimental mathematician's point of view
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Math. Soc.
2012
|
| Series: | Student mathematical library
65 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 504 S. graph. Darst. |
| ISBN: | 9780821887950 |
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MARC
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| 100 | 1 | |a Moll, Victor H. |d 1956- |e Verfasser |0 (DE-588)173099572 |4 aut | |
| 245 | 1 | 0 | |a Numbers and functions |b from a classical-experimental mathematician's point of view |c Victor H. Moll |
| 264 | 1 | |a Providence, R.I. |b American Math. Soc. |c 2012 | |
| 300 | |a XXIII, 504 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Student mathematical library |v 65 | |
| 650 | 0 | 7 | |a Funktion |0 (DE-588)4195664-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Funktion |0 (DE-588)4195664-3 |D s |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-025419485 | ||
Record in the Search Index
| _version_ | 1804149677559382017 |
|---|---|
| adam_text | Contents
Preface
xiii
Chapter
1.
The Number Systems
1
§1.1.
The natural numbers
1
§1.2.
An automatic approach to finite sums
3
§1.3.
Elementary counting
8
§1.4.
The integers and divisibility
11
§1.5.
The Euclidean algorithm
12
§1.6.
Modular arithmetic
19
§1.7.
Prime numbers
20
§1.8.
The rational numbers
29
§1.9.
The set of real numbers
40
§1.10.
Fundamental sequences and completions
52
§1.11.
Complex numbers
54
Chapter
2.
Factorials and Binomial Coefficients
57
§2.1.
The definitions
57
§2.2.
A counting argument
60
§2.3.
The generating function of binomial coefficients
61
§2.4.
An extension of the binomial theorem to
noninteger
exponents
63
vn
viii Contents
§2.5.
Congruences for factorials and binomial coefficients
66
§2.6.
The prime factorization of n!
76
§2.7.
The central binomial coefficients
82
§2.8.
Bertrand s postulate
88
§2.9.
Some generating functions involving valuations
91
§2.10.
The asymptotics of factorials: Stirling s formula
94
§2.11.
The trinomial coefficients
96
Chapter
3.
The Fibonacci Numbers
105
§3.1.
Introduction
105
§3.2.
What do they count?
106
§3.3.
The generating function
108
§3.4.
A family of related numbers
110
§3.5.
Some arithmetical properties
112
§3.6.
Modular properties of Fibonacci numbers
120
§3.7.
Continued fractions of powers of Fibonacci quotients
123
§3.8.
Fibonacci polynomials
124
§3.9.
Series involving Fibonacci numbers
129
Chapter
4.
Polynomials
133
§4.1.
Introduction
133
§4.2.
Examples of polynomials
134
§4.3.
The division algorithm
139
§4.4.
Roots of polynomials
141
§4.5.
The fundamental theorem of algebra
144
§4.6.
The solution of polynomial equations
146
§4.7.
Cubic polynomials
151
§4.8.
Quartic polynomials
155
Chapter
5.
Binomial Sums
159
§5.1.
Introduction
159
§5.2.
Power sums
160
§5.3.
Moment sums
167
Contents ix
§5.4.
Recurrences for powers of binomials
171
§5.5.
Calkin s identity
173
Chapter
6.
Catalan Numbers
179
§6.1.
The placing of parentheses
179
§6.2.
A recurrence
179
§6.3.
The generating function
180
§6.4.
Arithmetical properties
186
§6.5.
An integral expression
190
Chapter
7.
The Stirling Numbers of the Second Kind
191
§7.1.
Introduction
191
§7.2.
A recurrence
192
§7.3.
An explicit formula
194
§7.4.
The valuations of Stirling numbers
197
Chapter
8.
Rational Functions
211
§8.1.
Introduction
211
§8.2.
The method of partial fractions
213
§8.3.
Rational generating functions
217
§8.4.
The operator point of view
219
§8.5.
A dynamical system
221
§8.6.
Sums of four squares
225
§8.7.
The integration of rational functions
229
§8.8.
Symbolic integration. The methods of Hermite and
Rothstein-Träger 237
Chapter
9.
Wallis
Formula
245
§9.1.
An experimental approach
245
§9.2.
A proof based on recurrences
247
§9.3.
A proof based on generating functions
248
§9.4.
A trigonometric version
249
§9.5.
An automatic proof
252
Contents
Chapter
10. Farey
Fractions
255
§10.1.
Introduction
255
§10.2.
Farey fractions and the Stern-Brocot tree
255
§10.3.
The distribution of denominators
262
Chapter
11.
The Exponential Function
269
§11.1.
Introduction
269
§11.2.
Elementary properties of the exponential function
272
§11.3.
The constant
e
274
§11.4.
The series representation of
e
275
§11.5.
Arithmetical properties of
e
278 :
§11.6.
Continued fractions connected to
e
284
§11.7.
Derangements: The presence of
e in
combinatorics
288
§11.8.
The natural logarithm
293:
§11.9.
The binary expansion of In
2 294
¡
§11.10.
The irrationality of In
2 295
§11.11.
Harmonic numbers
302
Chapter
12.
Trigonometric Functions
309-
§12.1.
Introduction
309 і
§12.2.
The notion of angle
309
§12.3.
Sine and cosine
311
§12.4.
The additional trigonometric functions
314
§12.5.
The addition theorem
320
§12.6.
Stirling s formula and
π
323
§12.7.
The continued fraction of
π
325
§12.8.
The digits of
π
in base
16 330
§12.9.
Special values of trigonometric functions
331
§12.10.
The roots of a cubic polynomial
335
§12.11.
A special trigonometric integral
339
§12.12.
The infinite product for
sina;
343
Contents xi
§12.13.
The irrationality of
тг
346
§12.14.
Arctangent sums and a dynamical system
349
Chapter
13.
Bernoulli Polynomials
355
§13.1.
Introduction
355
§13.2.
The exponential generating function
356
§13.3.
Elementary properties of Bernoulli numbers
357
§13.4.
Integrals involving Bernoulli polynomials
366
§13.5.
A relation to Stirling numbers
369
§13.6.
Arithmetic properties of Bernoulli numbers
371
§13.7.
The Euler-MacLaurin summation formula
376
§13.8.
Bernoulli numbers and solitons
383
§13.9.
The Giuga-Agoh conjectured criterion for primality
384
Chapter
14.
A Sample of Classical Polynomials: Legendre,
Chebyshev, and Hermite
387
§14.1.
Introduction
387
§14.2.
Legendre polynomials
387
§14.3.
Chebyshev polynomials
400
§14.4.
Hermite polynomials
403
Chapter
15. Landen
Transformations
411
§15.1.
Introduction
411
§15.2.
An elementary example
413
§15.3.
The case of rational integrands
415
§15.4.
The evaluation of a quartic integral
417
§15.5.
An integrand of degree six
435
§15.6.
The original elliptic case
438
Chapter
16.
Three Special Functions:
Γ,φ,
and
ζ
445
§16.1.
Introduction
445
§16.2.
The gamma function
446
xii Contents
§16.3.
Elementary properties of the gamma function
447
§16.4.
Special values of the gamma function
449
§16.5.
The infinite product for the gamma function
450
§16.6.
The beta function
455
§16.7.
The
digamma
function
458
§16.8.
The Riemann
zeta
function
460
§16.9.
The values of
ζ(2η)
461
§16.10.
Apery s constant C(3)
463
Bibliography
473
Index
493
|
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| author | Moll, Victor H. 1956- |
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| author_facet | Moll, Victor H. 1956- |
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| author_sort | Moll, Victor H. 1956- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.25 |
| dewey-search | 515/.25 |
| dewey-sort | 3515 225 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T00:26:50Z |
| institution | BVB |
| isbn | 9780821887950 |
| language | English |
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| physical | XXIII, 504 S. graph. Darst. |
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| spelling | Moll, Victor H. 1956- Verfasser (DE-588)173099572 aut Numbers and functions from a classical-experimental mathematician's point of view Victor H. Moll Providence, R.I. American Math. Soc. 2012 XXIII, 504 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 65 Funktion (DE-588)4195664-3 gnd rswk-swf Funktion (DE-588)4195664-3 s DE-604 Student mathematical library 65 (DE-604)BV013184751 65 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025419485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moll, Victor H. 1956- Numbers and functions from a classical-experimental mathematician's point of view Student mathematical library Funktion (DE-588)4195664-3 gnd |
| subject_GND | (DE-588)4195664-3 |
| title | Numbers and functions from a classical-experimental mathematician's point of view |
| title_auth | Numbers and functions from a classical-experimental mathematician's point of view |
| title_exact_search | Numbers and functions from a classical-experimental mathematician's point of view |
| title_full | Numbers and functions from a classical-experimental mathematician's point of view Victor H. Moll |
| title_fullStr | Numbers and functions from a classical-experimental mathematician's point of view Victor H. Moll |
| title_full_unstemmed | Numbers and functions from a classical-experimental mathematician's point of view Victor H. Moll |
| title_short | Numbers and functions |
| title_sort | numbers and functions from a classical experimental mathematician s point of view |
| title_sub | from a classical-experimental mathematician's point of view |
| topic | Funktion (DE-588)4195664-3 gnd |
| topic_facet | Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025419485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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