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El. knyga: Ramanujan Summation of Divergent Series

  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2185
  • Išleidimo metai: 12-Sep-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319636306
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Ramanujan Summation of Divergent SeriesDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Lecture Notes in Mathematics 2185
  • Išleidimo metai: 12-Sep-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319636306
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The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.
1 Ramanujan Summation
1(30)
1.1 The Euler-MacLaurin Summation Formula
3(6)
1.2 Ramanujan's Constant of a Series
9(1)
1.3 A Difference Equation
9(6)
1.3.1 The Functions φf and Rf
10(5)
1.3.2 The Fundamental Theorem
15(1)
1.4 The Summation
15(16)
1.4.1 Definition and Examples
23(4)
1.4.2 The Fractional Sum
27(4)
1.4.3 Relation to Usual Summation
31(1)
2 Properties of the Ramanujan Summation
31(30)
2.1 Some Elementary Properties
31(10)
2.1.1 The Unusual Property of the Shift
31(6)
2.1.2 Summation by Parts
37(1)
2.1.3 Sums of Products
38(3)
2.2 Summation and Derivation
41(4)
2.3 The Case of an Entire Function
45(10)
2.3.1 The Sum of an Entire Function
45(6)
2.3.2 An Expression of Catalan's Constant
51(1)
2.3.3 Some Trigonometric Series
52(3)
2.4 Functional Relations for Fractional Sums
55(6)
3 Dependence on a Parameter
61(52)
3.1 Analyticity with Respect to a Parameter
61(22)
3.1.1 The Theorem of Analyticity
61(9)
3.1.2 Analytic Continuation of Dirichlet Series
70(2)
3.1.3 The Zeta Function of the Laplacian on S2
72(3)
3.1.4 More Zeta Series
75(4)
3.1.5 A Modified Zeta Function
79(1)
3.1.6 The Sums ΣRn≤1 Nkφf(n)
80(3)
3.2 Integration with Respect to a Parameter
83(12)
3.2.1 Interchanging ΣRn≤1 Nkφf1
83(2)
3.2.2 The Functional Equation for Zeta
85(1)
3.2.3 The Case of a Laplace Transform
86(5)
3.2.4 Series Involving Zeta Values
91(4)
3.3 Double Sums
95(18)
3.3.1 Definitions and Properties
95(5)
3.3.2 Some Formulas for the Stieltjes Constant γ1
100(6)
3.3.3 A Simple Proof of a Formula of Ramanujan
106(3)
3.3.4 The Functional Relation for Eisenstein Function G2
109(4)
4 Transformation Formulas
113(44)
4.1 A Borel Summable Series
113(7)
4.1.1 A Formal Transform
113(2)
4.1.2 Borel Summation
115(2)
4.1.3 Borel Summability of Euler-MacLaurin Series
117(3)
4.2 A Convergent Expansion
120(15)
4.2.1 Bernoulli Numbers of Second Kind
120(1)
4.2.2 Newton Interpolation Formula
121(2)
4.2.3 Evaluation of Δnf(1)
123(5)
4.2.4 The Convergent Transformation Formula
128(7)
4.3 Summation of Alternating Series
135(22)
4.3.1 Euler Summation of Alternating Series
135(6)
4.3.2 Properties of the Summation
141(3)
4.3.3 Relation with the Ramanujan Summation
144(7)
4.3.4 Generalization
151(6)
5 An Algebraic View on the Summation of Series
157(18)
5.1 Introduction
157(2)
5.2 An Algebraic Formalism
159(4)
5.3 Properties of the General Summation
163(6)
5.3.1 The Linearity Property
163(1)
5.3.2 The Shift Property
164(1)
5.3.3 The Associated Algebraic Limit
165(2)
5.3.4 Sum of Products
167(2)
5.4 Examples
169(6)
A Euler-MacLaurin and Euler-Boole Formulas
175(6)
A.1 A Taylor Formula
175(2)
A.2 The Euler-MacLaurin Formula
177(1)
A.3 The Euler-Boole Formula
178(3)
B Ramanujan's Interpolation Formula and Carlson's Theorem
181(10)
Bibliography 191(2)
Index 193
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