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Softcover ISBN:  9781470477851 
Product Code:  CHEL/334.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470477905 
Product Code:  CHEL/334.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9781470477851 
eBook ISBN:  9781470477905 
Product Code:  CHEL/334.S.B 
List Price:  $134.00 $101.50 
AMS Member Price:  $120.60 $91.35 

Book DetailsAMS Chelsea PublishingVolume: 334; 1949; 396 ppMSC: Primary 01; 40
From the Preface by J. E. Littlewood: “All [Hardy's] books gave him some degree of pleasure, but this one, his last, was his favourite. When embarking on it he told me that he believed in its value (as well he might), and also that he looked forward to the task with enthusiasm. He had actually given lectures on the subject at intervals ever since his return to Cambridge in 1931, and he had at one time or another lectured on everything in the book except Chapter XIII [The EulerMacLaurin sum formula] ... [I]n the early years of the century the subject [Divergent Series], while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity.”

Table of Contents

Front Cover

PREFACE

NOTE

CONTENTS

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 zetafunction

2.2. The functional equations for ζ(s), 𝓃(s), and L(s).

2.3. Euler's verification.

B. Euler and the series ll!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 11 + 1... by special Abelian methods.

4.11. Lindelof's and MittagLeffler'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 11+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 nonintegral 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 seriesΣnbeAina.

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 Σn1ic

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 overconvergence.

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 1505.

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. QuasiHausdorff transformations.

11.20. Regularity of a quasiHausdorff 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. Onesided 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 EULERMACLAURIN 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 EulerMaclaurin sum formula.

13.6. Limits as n ➔ ∞.

13.7. The sign and magnitude of the remainder term.

13.8. Poisson's proof of the EulerMaclaurin formula.

13.9. A formula of Fourier.

13.10. The casef(x) = xs and the Riemann zetafunction.

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 EulerMaclaurin formula by complex integration.

13.15. Summability of the EulerMaclaurin 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

Back Cover


Additional Material

Reviews

Review of 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


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From the Preface by J. E. Littlewood: “All [Hardy's] books gave him some degree of pleasure, but this one, his last, was his favourite. When embarking on it he told me that he believed in its value (as well he might), and also that he looked forward to the task with enthusiasm. He had actually given lectures on the subject at intervals ever since his return to Cambridge in 1931, and he had at one time or another lectured on everything in the book except Chapter XIII [The EulerMacLaurin sum formula] ... [I]n the early years of the century the subject [Divergent Series], while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity.”

Front Cover

PREFACE

NOTE

CONTENTS

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 zetafunction

2.2. The functional equations for ζ(s), 𝓃(s), and L(s).

2.3. Euler's verification.

B. Euler and the series ll!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 11 + 1... by special Abelian methods.

4.11. Lindelof's and MittagLeffler'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 11+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 nonintegral 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 seriesΣnbeAina.

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 Σn1ic

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 overconvergence.

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 1505.

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. QuasiHausdorff transformations.

11.20. Regularity of a quasiHausdorff 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. Onesided 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 EULERMACLAURIN 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 EulerMaclaurin sum formula.

13.6. Limits as n ➔ ∞.

13.7. The sign and magnitude of the remainder term.

13.8. Poisson's proof of the EulerMaclaurin formula.

13.9. A formula of Fourier.

13.10. The casef(x) = xs and the Riemann zetafunction.

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 EulerMaclaurin formula by complex integration.

13.15. Summability of the EulerMaclaurin 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

Back Cover

Review of 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