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E-raamat: Infinite Series in a History of Analysis: Stages Up to the Verge of Summability [De Gruyter e-raamatud]

  • Formaat: 142 pages, 23 Illustrations, black and white
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 26-Sep-2014
  • Kirjastus: De Gruyter Oldenbourg
  • ISBN-13: 9783110359831
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  • De Gruyter e-raamatud
  • Hind: 360,00 €*
  • * hind, mis tagab piiramatu üheaegsete kasutajate arvuga ligipääsu piiramatuks ajaks
Infinite Series in a History of Analysis: Stages Up to the Verge of SummabilitySuurem pilt
  • Formaat: 142 pages, 23 Illustrations, black and white
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 26-Sep-2014
  • Kirjastus: De Gruyter Oldenbourg
  • ISBN-13: 9783110359831
Teised raamatud teemal:
Wiley OnlineRohkem infot De Gruyter e-raamatute kohta

Für die Entwicklungsgeschichte der Analysis sind die unendlichen Reihen nicht nur ein roter Faden; seit dem 17. Jahrhundert untrennbar mit der Infinitesimalrechnung verbunden, bilden sie geradezu ihr Rückgrat. In Struktur und Anschaulichkeit sind sie ein dankbares Objekt der Didaktik. Rückblicke, Einsichten, Ausblicke sind geeignet, das Verständnis für den Lehrstoff "Analysis" zu vertiefen.



"Higher mathematics" once pointed towards the involvement of infinity. This we label analysis. The ancient Greeks had helped it to a first high point when they mastered the infinite. The book traces the history of analysis along the risky route of serial procedures through antiquity. It took quite long for this type of mathematics to revive in our region. When and where it did, infinite series proved the driving force. Not until a good two millennia had gone by, would analysis head towards Greek rigor again. To follow all that trial, error and final accomplishment, is more than studying history: It provides touching, worthwhile access to advanced calculus. Moreover, some steps beyond convergence show infinite series to naturally fit a wider frame.

Foreword vii
1 The Greek era: Speculation, ideas and the comet of rigor
1(10)
1.1 Infinity, challenging Greek philosophers
1(1)
1.2 Infinity, mastered by a Greek "geometer": Eudoxus
2(3)
1.3 Archimedes and the geometric series to be
5(4)
1.4 Memorizing a joint venture
9(2)
2 Successful experiments on infinite summation
11(8)
2.1 Oresmus and the geometric series
11(2)
2.2 Freestyle: Cavalieri summing indivisibles
13(3)
2.3 Leibniz' telescopes
16(3)
3 Series of functions: the prototypes
19(48)
3.1 Preliminary: Aspects
19(4)
3.1.1 Series versus sequences
19(1)
3.1.2 Series: evaluation and expansion
20(3)
3.2 The power of power series
23(23)
3.2.1 Nilakantha and the arcustangent series
23(3)
3.2.2 The hyperbolic logarithm and its series
26(3)
3.2.3 Newton and the binomial series
29(3)
3.2.4 Taylor series and expansion; ordinary power series
32(4)
3.2.5 The binomial series in the hands of Euler
36(6)
3.2.6 Euler's algebraical analysis: evaluation and valuation
42(4)
3.3 The power of trigonometric series
46(9)
3.3.1 Facing antipodes
46(1)
3.3.2 Euler evaluating a trigonometric series
46(3)
3.3.3 Priming problems
49(1)
3.3.4 The mysterious ... Fourier series
50(5)
3.4 Cauchy
55(12)
3.4.1 Series of powers in the hands of Cauchy and Laurent
55(6)
3.4.2 The Cauchy product
61(6)
4 Series seriously: the Greek comet reappears
67(22)
4.1 Rigor in retrospect and prospect
67(2)
4.2 Farewell to the infinitely small; the "epsilontics"
69(9)
4.2.1 Convergence, continuity
69(2)
4.2.2 Dirichlet, Heine: uniform continuity
71(1)
4.2.3 From Abel to Weierstraß: uniform convergence
72(6)
4.3 Welcome to irrationals: the complete space of real numbers
78(6)
4.4 Another complete space: the home of convergent sequences
84(5)
5 On the verge of summability
89(22)
5.1 Divergent series: suspected and respected
89(1)
5.2 The initiation of Cesaro and Abel summation
90(8)
5.2.1 Grandi's series, recurrent series
90(2)
5.2.2 Frobenius' theorem and the limit theorems of Cauchy and Abel
92(2)
5.2.3 The Cauchy product revisited by Abel and Cesaro
94(1)
5.2.4 Introducing the methods of basic Cesaro and of Abel means
95(3)
5.3 Features of Cesaro and Abel means
98(13)
5.3.1 Inclusion, limitation, efficiency
98(3)
5.3.2 The C1 limit: a continuous functional on a Banach space
101(2)
5.3.3 An early triumph of arithmetic means: Fejer's Theorem
103(1)
5.3.4 On the scale of Cesaro means
103(3)
5.3.5 Cesaro's Cauchy product
106(1)
5.3.6 Inverse theorems, Tauber's theorem
107(4)
Appendixes (of notes, proofs, exercises) 111(8)
Life Data 119(2)
References 121(6)
Index 127
Hans-Heinrich Körle, form.University of Marburg, Germany.
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