Divergent Series

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
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