Divergent Series

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Title page 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), η(s), and L(s) 2.3. Euler's verification B. Euler and the series 1 — 1!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 Sum(φ^{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. Nörlund means 4.2. Regularity and consistency of Nörlund 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. Lindelöfs 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. Hölder's means 5.3. Simple theorems concerning Hölder summability 5.4. Cesàro means 5.5. Means of non-integral order 5.6. A theorem concerning integral resultants 5.7. Simple theorems concerning Cesàro summability 5.8. The equivalence theorem 5.9. Mercer's theorem and Schur's proof of the equivalence 5.10. Other proofs of Mercer's theorem 5.11. Infinite limits 5.12. Cesàro and Abel summability 5.13. Cesàro means as Nörlund 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 {n².α} NOTES ON CHAPTER V VI. ARITHMETIC MEANS (2) 6.1. Tauberian theorems for Cesàro 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)^(-s) 6.7. Another condition for summability 6.8. Integrals 6.9. The binomial series 6.10. The series etc 6.11. The case β = -1 6.12. The series etc 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 96 a 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 Sum(n^(-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 X. 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 Sum(z^n) 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 Hölder and Cesàro 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 Hölder and Cesàro means 11.12. Some special cases 11.13. Logarithmic cases 11.14. Exponential cases 11.15. The Legendre series for χ(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 concerning 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. Vijayaraghavan'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 φ_n(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 tho Euler-Maclaurin formula 13.9. A formula of Fourier 13.10. The case f(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 C 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 sries 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 LIST OF DEFINITIONS GENERAL INDEX