Divergent Series

Godfrey H. Hardy (Autor)

Buch | Softcover

396 Seiten

1949
American Mathematical Society (Verlag)
978-1-4704-7785-1 (ISBN)

Infos

Review of the original edition: “This is an inspiring textbook for students who know the theory of functions of real and complex variables and wish further knowledge of mathematical analysis. There are no problems displayed and labelled ""problems,"" but one who follows all of the arguments and calculations of the text will find use for his ingenuity and pencil. The book deals with interesting and important problems and topics in many fields of mathematical analysis, to an extent very much greater than that indicated by the titles of the chapters. It is, of course, an indispensable handbook for those interested in divergent series. It assembles a considerable part of the theory of divergent series, which has previously existed only in periodical literature. Hardy has greatly simplified and improved many theories, theorems and proofs. In addition, numerous acknowledgements show that the book incorporates many previously unpublished results and improvements of old results, communicated to Hardy by his colleagues and by others interested in the book.” - Mathematical Reviews

Inhalt

NOTE ON CONVENTIONS
I INTRODUCTION
1.1. The sum of a series.
1.2. Some calculations with divergent series.
1.3. First definitions.
1.4. Regularity of a method.
1.5. Divergent integrals and generalized limits of functions of a continuous variable.
1.6. Some historical remarks.
1.7. A note on the British analysts of the early nineteenth century.
NOTES ON CHAPTER I
II SOME HISTORICAL EXAMPLES
2.1. Introduction.
A. Euler and the functional equation of Riemann's zeta-function
2.2. The functional equations for ζ(s), 𝓃(s), and L(s).
2.3. Euler's verification.
B. Euler and the series l-l!x+2!x2- •••
2.4. Summation of the series.
2.5. The asymptotic nature of the series.
2.6. Numerical computations.
C. Fourier and Fourier's theorem
2.7. Fourier's theorem.
2.8. Fourier's first formula for the coefficients.
2.9. Other forms of the coefficients and the series.
2.10. The validity of Fourier's formulae.
D. Heaviside's exponential series
2.11. Heaviside on divergent series.
2.12. The generalized exponential series.
2.13. The series ΣΦ(r)(x).
2.14. The generalized binomial series.
NOTES ON CHAPTER II
III GENERAL THEOREMS
3.1. Generalities concerning linear transformations.
3.2. Regular transformations.
3.3. Proof of Theorems 1 and 2.
3.4. Proof of Theorem 3.
3.5. Variants and analogues.
3.6. Positive transformations.
3.7. Knopp's kernel theorem.
3.8. An application of Theorem 2.
3.9. Dilution of series.
NOTES ON CHAPTER III
IV SPECIAL METHODS OF SUMMATION
4.1. Norlund means.
4.2. Regularity and consistency of Norlund means.
4.3. Inclusion.
4.4. Equivalence.
4.5. Another theorem concerning inclusion.
4.6. Euler means.
4.7. Abelian means.
4.8. A theorem of inclusion for Abelian means.
4.9. Complex methods.
4.10. Summability of 1-1 + 1-... by special Abelian methods.
4.11. Lindelof's and Mittag-Leffler's methods.
4.12. Means defined by integral functions.
4.13. Moment constant methods.
4.14. A theorem of consistency.
4.15. Methods ineffective for the series 1-1+1-....
4.16. Riesz's typical means.
4.17. Methods suggested by the theory of Fourier series.
4.18. A general principle.
NOTES ON CHAPTER IV
V ARITHMETIC MEANS ( 1)
5.1. Introduction.
5.2. Holder's means.
5.3. Simple theorems concerning H61der summability.
5.4. Cesaro means.
5.5. Means of non-integral order.
5.6. A theorem concerning integral resultants.
5.7. Simple theorems concerning Cesaro summability.
5.8. The equivalence theorem.
5.9. Mercer's theorem and Schur's proof of the equivalence theorem.
5.10. Other proofs of Mercer's theorem.
5.11. Infinite limits.
5.12. Cesaro and Abel summability.
5.13. Cesaro means as Norlund means.
5.14. Integrals.
5.15. Theorems concerning summable integrals.
5.16. Riesz's arithmetic means.
5.17. Uniformly distributed sequences.
5.18. The uniform distribution of {n2α}.
NOTES ON CHAPTER V
VI ARITHMETIC MEANS (2)
6.1. Tauberian theorems for Cesaro summability.
6.2. Slowly oscillating and slowly decreasing functions.
6.3. Another Tauberian condition.
6.4. Convexity theorems.
6.5. Convergence factors.
6.6. The factor (n+ 1)-8 •
6.7. Another condition for summability.
6.8. Integrals.
6.9. The binomial series.
6.10. The series Σnαeniθ.
6.11. The case ß
= - I.
6.12. The seriesΣn-beAina.
NOTES ON CHAPTER VI
VII TAUBERIAN THEOREMS FOR POWER SERIES
7.1. Abelian and Tauberian theorems.
7.2. Tauber's first theorem.
7.3. Tauber's second theorem.
7.4. Applications to general Dirichlet's series.
7.5. The deeper Tauberian theorems.
7.6. Proof of Theorems 96 and 96a.
7.7. Proof of Theorems 91 and 91 a.
7.8. Further remarks on the relations between the theorems of§
7.5.
7.9. The series Σn-1-ic
7.10. Slowly oscillating and slowly decreasing functions.
7.11. Another generalization of Theorem 98.
7.12. The method of Hardy and Littlewood.
7.13. The 'high indices'
theorem.
NOTES ON CHAPTER VII
VIII THE METHODS OF EULER AND BOREL (1)
8.1. Introduction.
8.2. The (E, q) method.
8.3. Simple properties of the (E, q) method.
8.4. The formal relations between Euler's and Borel's methods.
8.5. Borel's methods.
8.6. Normal, absolute, and regular summability.
8.7. Abelian theorems for Borel summability.
8.8. Analytic continuation of a function regular at the origin: the polygon of summability.
8.9. Series representing functions with a singular point at the origin.
8.10. Analytic continuation by other methods.
8.11. The summability of certain asymptotic series.
NOTES ON CHAPTER VIII
IX THE METHODS OF EULER AND BOREL (2)
9.1. Some elementary lemmas.
9.2. Proof of Theorem 137.
9.3. Proof of Theorem 139.
9.4. Another elementary lemma.
9.5. Ostrowski's theorem on over-convergence.
9.6. Tauberian theorems for Borel summability.
9.7. Tauberian theorems (continued).
9.8. Examples of series not summable (B).
9.9. A theorem in the opposite direction.
9.10. The (e, c) method of summation.
9.11. The circle method of summation.
9.12. Further remarks on Theorems 150-5.
9.13. The principal Tauberian theorem.
9.14. Generalizations.
9.15. The series Σzn.
9.16. Valiron's methods.
NOTES ON CHAPTER IX
X MULTIPLICATION OF SERIES
10.1. Formal rules for multiplication.
10.2. The classical theorems for multiplication by Cauchy's rule.
10.3. Multiplication of summable series.
10.4. Another theorem concerning convergence.
10.5. Further applications of Theorem 170.
10.6. Alternating series.
10.7. Formal multiplication.
10.8. Multiplication of integrals.
10.9. Euler summability.
10.10. Borel summability.
10.11. Dirichlet multiplication.
10.12. Series infinite in both directions.
10.13. The analogues of Cauchy's and Mertens's theorems.
10.14. Further theorems.
10.15. The analogue of Abel's theorem.
NOTES ON CHAPTER X
XI HAUSDORFF MEANS
11.1. The transformation 𝛿.
11.2. Expression of the (E, q) and (C, 1) transformations in terms of 𝛿.
11.3. Hausdorff's general transformation.
11.4. The general H6lder and Cesaro transformations as 𝕳
transformations.
11.5. Conditions for the regularity of a real Hausdorff transformation.
11.6. Totally monotone sequences.
11.7. Final form of the conditions for regularity.
11.8. Moment constants.
11.9. Hausdorff's theorem.
11.10. Inclusion and equivalence of 𝕳
methods.
11.11. Mercer's theorem and the equivalence theorem for Holder and Cesaro means.
11.12. Some special cases
11.13. Logarithmic cases.
11.14. Exponential cases.
11.15. The Legendre series for x(x).
11.16. The moment constants of functions of particular classes.
11.17. An inequality for Hausdorff means.
11.18. Continuous transformations.
11.19. Quasi-Hausdorff transformations.
11.20. Regularity of a quasi-Hausdorff transformation.
11.21. Examples.
NOTES ON CHAPTER XI
XII WIENER'S TAUBERIAN THEOREMS
12.1. Introduction.
12.2. Wiener's condition.
12.3. Lemmas concernin~ Fourier transforms.
12.4. Lemmas concerning the class U.
12.5. Final lemmas.
12.6. Proof of Theorems 221 and 220.
12.7. Wiener's second theorem.
12.8. Theorems for the interval (0, ∞).
12.9. Some special kernels.
12.10. Application of the general theorems to some special kernels.
12.11. Applications to the theory of primes.
12.12. One-sided conditions.
12.13. Vijayara~havan's theorem.
12.14. Proof of Theorem 238.
12.15. Borel summability.
12.16. Summability (R, 2).
NOTES ON CHAPTER XII
XIII THE EULER-MACLAURIN SUM FORMULA
13.1. Introduction.
13.2. The Bernoullian numbers and functions.
13.3. The associated periodic functions.
13.4. The signs of the functions Φn(x).
13.5. The Euler-Maclaurin sum formula.
13.6. Limits as n ➔
∞.
13.7. The sign and magnitude of the remainder term.
13.8. Poisson's proof of the Euler-Maclaurin formula.
13.9. A formula of Fourier.
13.10. The casef(x) = x-s and the Riemann zeta-function.
13.11. The case f(x) = log(x+c) and Stirling's theorem.
13.12. Generalization of the formulae.
13.13. Other formulae for 0.
13.14. Investigation of the Euler-Maclaurin formula by complex integration.
13.15. Summability of the Euler-Maclaurin series.
13.16. Additional remarks.
13.17. The ℜ
definition of the sum of a divergent series.
NOTES ON CHAPTER XIII
APPENDIX I On the evaluation of certain definite integrals by means of divergent series
APPENDIX II The Fourier kernels of certain methods of summation
APPENDIX III On Riemann and Abel summability
APPENDIX IV On Lambert and Ingham summability
APPENDIX V Two theorems of M. L. Cartwright
LIST OF BOOKS
LIST OF PERIODICALS
LIST OF AUTHORS OF ORIGINAL PAPERS AND BOOKS NOT INCLUDED IN THE LIST OF BOOKS
LIST OF DEFINITIONS
GENERAL INDEX

Produkt-Details

• Erscheint lt. Verlag: 31.1.1949
• Reihe/Serie: AMS Chelsea Publishing ; 334
• Verlagsort: Providence
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-10: 1-4704-7785-8 / 1470477858
• ISBN-13: 978-1-4704-7785-1 / 9781470477851
• Zustand: Neuware