An introduction to the theory of infinite series:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Delhi
Kalpaz
2017
|
| Ausgabe: | Indian reprint |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | "First published in 1908"--Title page verso |
| Beschreibung: | xiii, 511 pages illustrations (black and white) 21 cm |
| ISBN: | 9789351285946 9351285944 |
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| 245 | 1 | 0 | |a An introduction to the theory of infinite series |c by T.J.I'A. Bromwich, M.A.,F.R.S. |
| 250 | |a Indian reprint | ||
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Datensatz im Suchindex
| _version_ | 1804177833114730496 |
|---|---|
| adam_text | CONTENTS.
CHAPTER I.
FAOn
Sequences and Limits, ..... 1-16
.*•
Convergence of Sequences. Monotonic Sequence*. General
Principle of Convergence. Upper and Lower Limits.
Maximum and Minimum Limits. Sum of an Infinite Series.
0
Examples, ....... 17
CHAPTER II.
Series of Positive Terms,...........................22-42
Cauchy’» Condensation Test. Comparison Test. Integral
Test. Logarithmic Scale. Ratio Testa Ermskoff’s Tests.
Another sequence of Tests. Notes on Tests of Convergence.
Examples, ....... 42
CHAPTER III.
Series in General,..................................46-59
Absolute and Non-Absolute Convergence. Testa of Abel
and Dirichlet. Alternate Series; Ratio Test. Abel’s
Lemma. Euler s Transformation.
Examples, ....... 60
CHAPTER IV.
Absolute Convergence,...............................63-70
Derangement. Applications. Riemann’s and Pringaheim’s
Theorems.
~RyA.arpr.itR,
70
CONTENTS.
CHAPTER V.
Double Series, - - - -
Sum of a Double Series. Repeated Summation. Series of
Positive Terms. Tests for Convergence. Absolute Con-
vergence. Multiplication of Series. Mortens* and Prings-
heim’s Theorems. Substitution of Power-Series in another
Power-Series. N on - Absolute Convergen oe.
Examples, -------
Theta-Series (16) ; other Elliptic Function-Series (19-24).
CHAPTER VI.
Infinite Products, -
Wei era trass’s Inequalities. Positive Terms. Terms of
either Sign. Absolute Convergence. Gamma-Product.
Examples, -
Theta-Products (14-20).
CHAPTER VII.
Series of Variable Teems, ----- 108-125
Uniform Convergence of Sequences. Uniform Convergence
of Series. Weierstraas’s, Abel’s, and Dirichlet’s Tests.
Continuity, Integration, and Differentiation of Series.
Products. Tannery s Theorems.
Examples, - - 125
Bendixson’s Test of Uniform Convergence (14).
CHAPTER VIII.
Power Series, ------- 126-142
Intervals of Convergence. Abel’s Theorem. Continuity,
Differentiation, Integration. Theorem of Identical Equality
between two Power-Series. Multiplication and Division.
Reversion of Series. Lagrange’s Series.
Special Power Series, ------ 143-160
Exponential Series. Sine and Cosine Series. Binomial
Series. Logarithmic Series, arc sin and arc tan Series.
2r* cos «0, Sr*1 sin n0, 2^»* cos «0, S-f* sin »0.
n n
Examples, -
Series for rr (A. 48); Abel’s Theorem (B. 13-21); Lagrange’s
Series (B. 30, 31) ; Differential Equations (B. 34-36).
72-90
90
95-103
103
161
CONTENTS.
XI
CHAPTER IX.
Trigonometrical Investigations, -
Sines and Cosines of Multiple Angles. The Sine and
Cosine Products. Hie Cotangent Series.
Examples, - - - - -
CHAPTER X.
Complex Series and Products, -
Complex Numbers. De Moivre’a Theorem. Convergence
of Complex Sequences. Absolute Convergence of Series
and Products. Pringsheira s Ratio Testa for Absolute
Convergence. Weierstrasa’s Test for Power-Series. Abel s
and Diricblet s Tests for Convergence. Uniform Con-
vergence. Circle of Convergence. Abel s Theorem.
Poisson s Integral. Taylor s Theorem. Exponential Series.
Sine and Cosine Series. Logarithmic Series; arc sin and
arc tan Series. Binomial Series. Differentiation of Trigo-
nometrical Series. Sine and Cosine Products. The Cotan-
gent Series. Bernoulli s Numbers. Bernoulli n Functions.
Euler s Summation Formula.
Examples, -
Triflection of an Angle (A. 12); Abel’s Theorem (B. 27,
C. 9-11); Weierstraas’s Double Series Theorem (B. 32).
CHAPTER XI.
Non-Convebgknt and Asymptotic Series,
Bibliography. Historical Introduction. General Con-
siderations.
Borel s Method of Summation, -
BoreFs Integral. Condition of Consistency. Addition of
Terms to a Sununable Series. Examples of Summation.
Absolute Sumiuabilifcy. Multiplication. Continuity, Differ-
entiation, and Integration. Analogue of Abel s Theorem.
Suxnw ble Power-Series.
Other Methods of Summation, -
Borel s and Le Roy s other Definitions. An Extension of
Borel’s Definition. Eulei^s Series and BorePa Integral.
Examples of Euler’s Series. Ceskrota Mean. Extension of
Frobenius’a Theorem. Multiplication of Series. Examples
of Oeskro s Method. Borel s Sum and Ceskro’s Mean.
rAOflB
177-187
•
188
192-240
241
261-267
267-296
297-322
CONTENTS.
Mon
Asymptotic Series, - 322-346
Euler’s use of Asymptotic Series. Remainder in Euler’s
Formula. Logarithmic Integral. Fresnel’s Integrals.
Stirling’s Series. Poincaré’s Theory of Asymptotic Series.
Stokes’ Asymptotic Formula. Summation of Asymptotic
Series. Applications to Differential Equations.
Examples, - .... 347
Fejéris and de la Vallée Poussin’s Theorems (5-7).
*
APPENDIX I.
Arithmetic Theory of Irrational Numbers and Limits, 357-389
Infinite Decimals. Dedekind’s Definition. Algebraic Oper-
ations with Irrational Numbers. Monotonie Sequences.
Extreme limits of a Sequence. General Principle of Con-
vergence. Limits of Quotients. Extension of Abel’s
Lemma. Theorems on Limits.
Examples, ------- 390
Infinite Sets (15-17) ; Goursat’s Lemma (18-20) ; Continuous
Functions (21-23).
APPENDIX II.
Definitions of the Logarithmic and Exponential
Functions,.................................... 396-410
Definition and Fundamental Properties of the Logarithm.
Exponential Function. Logarithmic Scale of Infinity.
Exponential Series. Arithmetic Definition of an Integral
(Single and Double).
Examples, - 410
APPENDIX III.
Some Theorems on Infinite Integrals and Gamma-
Functions, ...................................414-466
Convergence, Divergence, and Oscillation. Definition as a
Limit of a Sum. Tests of Convergence. Analogue of Abel’s
Lemma (or Second Theorem of the Mean). Absolute
CONTENTS.
xiií
Convergence. Abel’s and Dirich let’s Tests. Frullani’s
Integrals. Uniform Convergence. Weierstrasa’a, Abel’s
and Dirichiefs Tests for Uniform Convergence. Continuity,
Differentiation, and Integration. Special Intégrala Limit*
ing Values of Intégrala Dirich let’s Intégrala Jordan’s
Integral. Integration of Seriea Inversion of Repeated
Intégrala The Gamma Integral. Stirling’s Asymptotic
Formula. Gamma-Function Formulae.
Examples, -
Gamma-Functions (85-47» 56-67).
EASY MISCELLANEOUS EXAMPLES.
Examples, -------
HARDER MISCELLANEOUS EXAMPLES.
Examples, -
Riemann s Discontinuous Series (18); Lagrange’s Series
(19,90); Weierstrass’s Non-Differentiable Function (39-86);
Riemann’s ¿-Function (44-49); Riemann’s Theorems on
Trigonometrical Series (64-66); Jacobi’s Theta-Functions
(70-77); Functions without Analytical Continuations (78-
85) ; Kumtner’s Series (86); Double Power-Series (87-100).
INDEX OF SPECIAL INTEGRALS, PRODUCTS AND
SERIES,......................................
PACKS
467
479-483
484-506
507-508
GENERAL INDEX,
509-511
AN INTRODUCTION TO THE THEORY OF
INFINITE SERIES
—T. J. I a. BROMWICH
This book is based on courses oflectures on Elementary Analysis given at Queen s
College, Galway, during each of the sessions 1902-1907. But additions have
naturally been made in preparing the manuscript. In Particular the whole of
chapter XI and greater part of the appendices have been added. This historic book
may have numerous typos and missing text. In Chapter 1., a preliminary account
is given of the notions of a limit and of convergence. I have not in this chapter
attempted to supply arithmetic proofs of the fundamental theorems concerning
the existence of limits, but have allowed their truth to rest on an appeal to the
readers intuition, in the hope that the discussion may thus be made more
attractive to beginners. An arithmetic treatment will be found in Appendix-1.,
where Dedekind s definition of irrational numbers is adopted as fundamental,
this method leads at once the monotonic principle of convergence, from which
the existence limits is deduced.
J
Thomas John I Anson Bromwich was born on 8 February 1875, in
Wolverhampton, England. He was descended from Bryan I’Anson, of Ashby St.
Legers, Sheriff of London and father of the 17th century 1st Baronet Sir Bryan
I’Anson of Bassctbury. His parents emigrated to South Africa, where in 1892 he
graduated from high school. He attended St John’s College, Cambridge, where in
1895 he became Senior Wrangler. In 1897, he became a lecturer at St. John’s.
From 1902 to 1907, he was a Professor of mathematics at Queen’s College,
Galway. In 1906, he was elected a Fellow of the Royal Society. In 1907, he
returned to Cambridge and again became a Fellow and lecturer at St. John’s. He
was a vice president of the Royal Society in 1919 and 1920. He died in
Northampton on 24 August 1929.
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| discipline | Mathematik |
| edition | Indian reprint |
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| spelling | Bromwich, Thomas John I'Anson 1875-1929 Verfasser (DE-588)116685239 aut An introduction to the theory of infinite series by T.J.I'A. Bromwich, M.A.,F.R.S. Indian reprint Delhi Kalpaz 2017 xiii, 511 pages illustrations (black and white) 21 cm txt rdacontent n rdamedia nc rdacarrier "First published in 1908"--Title page verso Reihe (DE-588)4049197-3 gnd rswk-swf Series, Infinite Reihe (DE-588)4049197-3 s 1\p DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029889357&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029889357&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
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| title | An introduction to the theory of infinite series |
| title_auth | An introduction to the theory of infinite series |
| title_exact_search | An introduction to the theory of infinite series |
| title_full | An introduction to the theory of infinite series by T.J.I'A. Bromwich, M.A.,F.R.S. |
| title_fullStr | An introduction to the theory of infinite series by T.J.I'A. Bromwich, M.A.,F.R.S. |
| title_full_unstemmed | An introduction to the theory of infinite series by T.J.I'A. Bromwich, M.A.,F.R.S. |
| title_short | An introduction to the theory of infinite series |
| title_sort | an introduction to the theory of infinite series |
| topic | Reihe (DE-588)4049197-3 gnd |
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