Summability Methods

Summability methodsFrom Wikipedia, the free encyclopediaContents1 Abels summation formula 11.1 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 EulerMascheroni constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Representation of Riemanns zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Reciprocal of Riemann zeta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Abels theorem 32.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Outline of proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Abelian and tauberian theorems 63.1 Abelian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Tauberian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 AbelPlana formula 84.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 BochnerRiesz mean 105.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS6 Borel summation 126.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.1.1 Borels exponential summation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.1.2 Borels integral summation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.1.3 Borels integral summation method with analytic continuation . . . . . . . . . . . . . . . . 136.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.2 Equivalence of Borel and weak Borel summation . . . . . . . . . . . . . . . . . . . . . . . 136.2.3 Relationship to other summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 Uniqueness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.1 Watsons theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.2 Carlemans theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.1 The geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.2 An alternating factorial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.3 An example in which equivalence fails . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 Existence results and the domain of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5.1 Summability on chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5.2 The Borel polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5.3 A Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Cauchy principal value 207.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2 Distribution theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2.1 Well-denedness as a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2.2 More general denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Darbouxs formula 248.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24CONTENTS iii9 Dimensional regularization 259.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610Divergent series 2710.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.2Theorems on methods for summing divergent series . . . . . . . . . . . . . . . . . . . . . . . . . 2810.3Properties of summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.4Classical summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.4.1 Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.4.2 Sum of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5Nrlund means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5.1 Cesro summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6Abelian means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6.1 Abel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.6.2 Lindelf summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7.1 Analytic continuation of power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7.2 Euler summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.7.3 Analytic continuation of Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.7.4 Zeta function regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.8Integral function means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.8.1 Borel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.8.2 Valirons method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.9Moment methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.9.1 Borel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10Miscellaneous methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10.1 Hausdor transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10.2 Hlder summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10.3 Huttons method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10.4 Ingham summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.5 Lambert summability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.6 Le Roy summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.7 Mittag-Leer summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.8 Ramanujan summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.9 Riemann summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.10Riesz means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.10.11Valle-Poussin summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36iv CONTENTS11Euler summation 3711.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812EulerMaclaurin formula 3912.1The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.1.1 The Bernoulli polynomials and periodic function. . . . . . . . . . . . . . . . . . . . . . . 4012.1.2 The remainder term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.3 Applicable formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.2Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.1 The Basel problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.2 Sums involving a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.3 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.4 Asymptotic expansion of sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.3Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.3.1 Derivation by mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413Hadamard regularization 4513.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514HardyLittlewood tauberian theorem 4714.1Statement of the theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.1 Series formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.2 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.2Karamatas proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3.1 Non-positive coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3.2 Littlewoods extension of Taubers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3.3 Prime number theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015Hlder summation 5115.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116Lambert summation 52CONTENTS v16.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217Littlewoods Tauberian theorem 5317.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318Mittag-Leer summation 5518.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619Nachbins theorem 5719.1Exponential type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.2 type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.3Borel transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.4Nachbin resummation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.5Frchet space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5920Perrons formula 6020.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6020.2Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6020.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6020.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6121Poisson summation formula 6221.1Forms of the equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.2Distributional formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.3Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6321.4Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6321.5Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5.1 Method of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5.3 Ewald summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5.4 Lattice points in a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6421.5.5 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.6Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65vi CONTENTS21.6.1 Selberg trace formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6521.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6621.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6621.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6621.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622Ramanujan summation 6722.1Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6722.2Sum of divergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6822.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923Regularization (physics) 7023.1Realistic regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.1.1 Conceptual problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.2 Paulis conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.3 Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.4 Minimal realistic regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.2Transport theoretic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224Resummation 7424.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.2Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7425Riesz mean 7525.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.2Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626Series acceleration 7726.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7726.2Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7726.3Eulers transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.4Conformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.5Non-linear sequence transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.5.1 Aitken method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7927SilvermanToeplitz theorem 8027.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8028Slowly varying function 81CONTENTS vii28.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8128.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8128.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8228.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8229Summation by parts 8329.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8329.2Newton series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8329.3Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8429.4Similarity with an integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8429.5Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8530Wieners tauberian theorem 8630.1The condition in L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8630.1.1 Tauberian reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8630.1.2 Discrete version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.2The condition in L2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8831WienerIkehara theorem 8931.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8931.2Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8931.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9032Zeta function regularization 9132.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9132.2Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.3Relation to other regularizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.4Relation to Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9232.5Heat kernel regularization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9332.6History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9432.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9432.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9432.9Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 9632.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9632.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9732.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Chapter 1Abels summation formulaAnother concept sometimes known by this name is summation by parts.In mathematics, Abels summation formula, introduced by Niels Henrik Abel, is intensively used in number theoryto compute series.1.1 IdentityLet anbe a sequence of real or complex numbers and (x) a function of class C1. Then1nxan(n) = A(x)(x) x1A(u)(u) duwhereA(x) :=1nxan .Indeed, this is integration by parts for a RiemannStieltjes integral.More generally, we havex1 . It may be used to derive Dirichlets theorem, that is, (s) has a simple pole withresidue 1 in s = 1.1.2.3 Reciprocal of Riemann zeta functionIf an= (n) is the Mbius function and (x) =1xs, then A(x) = M(x) =nx (n) is Mertens function and1(n)ns= s1M(u)u1+s du.This formula holds for (s) > 1 .1.3 See alsoSummation by parts1.4 ReferencesApostol, Tom(1976), Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag.Chapter 2Abels theoremThis article is about Abels theorem on power series. For Abels theorem on algebraic curves, see AbelJacobi map.For Abels theoremon the insolubility of the quintic equation, see AbelRuni theorem. For Abels theoremon lineardierential equations, see Abels identity. For Abels theorem on irreducible polynomials, see Abels irreducibilitytheorem.In mathematics, Abels theorem for power series relates a limit of a power series to the sum of its coecients. It isnamed after Norwegian mathematician Niels Henrik Abel.2.1 TheoremLet a = {ak: k 0} be any sequence of real or complex numbers and letGa(z) =k=0akzkbe the power series with coecients a. Suppose that the series k=0 akconverges. Thenlimz1Ga(z) =k=0ak, ()where the variable z is supposed to be real, or, more generally, to lie within any Stolz angle, that is, a region of theopen unit disk where|1 z| M(1 |z|)for some M. Without this restriction, the limit may fail to exist: for example, the power seriesn>0(z3nz23n)/nconverges to 0 at z=1, but is unbounded near any point of the form ei/3n, so the value at z=1 is not the limit as z tendsto 1 in the whole open disk.Note that Ga(z) is continuous on the real closed interval [0, t] for t < 1, by virtue of the uniform convergence of theseries on compact subsets of the disk of convergence.Abels theorem allows us to say more, namely that Ga(z) iscontinuous on [0, 1].34 CHAPTER 2. ABELS THEOREM2.2 RemarksAs an immediate consequence of this theorem, if z is any nonzero complex number for which the series k=0 akzkconverges, then it follows thatlimt1Ga(tz) =k=0akzkin which the limit is taken from below.The theorem can also be generalized to account for innite sums. Ifk=0ak= then the limit from below limz1 Ga(z) will tend to innity as well. However, if the series is only known to bedivergent, the theorem fails; take for example, the power series for11+z . The series is equal to 1 1 + 1 1 + at z= 1 , but 1/(1 + 1) = 1/2 .2.3 ApplicationsThe utility of Abels theorem is that it allows us to nd the limit of a power series as its argument (i.e.z ) approaches1 from below, even in cases where the radius of convergence,R , of the power series is equal to 1 and we cannotbe sure whether the limit should be nite or not. See e.g. the binomial series. Abels theorem allows us to evaluatemany series in closed form. For example, whenak=(1)k/(k+1) , we obtainGa(z) =ln(1+z)/z for0 0 we can take N large enough to make the initial segment of terms up to cNaverage to at most /2, while each term in the tail is bounded by /2 so that the average is also necessarily bounded.The name derives from Abels theorem on power series. In that case L is the radial limit (thought of within thecomplex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with termanznand set z = rei. That theorem has its main interest in the case that the power series has radius of convergence exactly1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] sothat the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum ofthe an. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless,if the sum of the an exists, it is equal to the limit over r. This therefore ts exactly into the abstract picture.3.2 Tauberian theoremsPartial converses to abelian theorems are called tauberian theorems. The original result of Tauber (1897) statedthat if we assume also63.3. REFERENCES 7an = o(1/n)(see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent.This was strengthened by J. E. Littlewood: we need only assume O(1/n).A sweeping generalization is the HardyLittlewood tauberian theorem.In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, andits values there are equal to those of the Lim functional. A tauberian theorem states, under some growth condition,that the domain of L is exactly the convergent sequences and no more.If one thinks of L as some generalised type of weighted average, taken to the limit, a tauberian theorem allows oneto discard the weighting, under the correct hypotheses. There are many applications of this kind of result in numbertheory, in particular in handling Dirichlet series.The development of the eld of tauberian theorems received a fresh turn with Norbert Wiener's very general results,namely Wieners tauberian theorem and its large collection of corollaries. The central theorem can now be proved byBanach algebra methods, and contains much, though not all, of the previous theory.3.3 ReferencesKorevaar, Jacob (2004). Tauberian theory. A century of developments. Grundlehren der MathematischenWissenschaften 329. Springer-Verlag. ISBN 978-3-540-21058-0.Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cam-bridge tracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press. pp. 147167. ISBN 0-521-84903-9.Tauber, A. (1897). Ein Satz aus der Theorie der unendlichen Reihen (A theorem from the theory of inniteseries)". Monatsh. F. Math. (in German) 8: 273277. doi:10.1007/BF01696278. JFM 28.0221.02.Wiener, N. (1932). Tauberian theorems. Annals of Mathematics33 (1): 1100. doi:10.2307/1968102.JSTOR 1968102.3.4 External linksHazewinkel, Michiel, ed. (2001), Tauberian theorems, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Chapter 4AbelPlana formulaIn mathematics, the AbelPlana formula is a summation formula discovered independently by Niels Henrik Abel(1823) and Giovanni Antonio Amedeo Plana (1820). It states thatn=0f(n) =0f(x) dx +12f(0) + i0f(it) f(it)e2t1dt.It holds for functions f that are holomorphic in the region Re(z) 0, and satisfy a suitable growth condition in thisregion; for example it is enough to assume that |f| is bounded by C/|z|1+ in this region for some constants C, > 0,though the formula also holds under much weaker bounds. (Olver 1997, p.290).An example is provided by the Hurwitz zeta function,(s, ) =n=01(n + )s=1ss 1+12s+ 20sin_s arctant_(2+ t2)s2dte2t1.Abel also gave the following variation for alternating sums:n=0(1)nf(n) =12f(0) + i0f(it) f(it)2 sinh(t)dt.4.1 See alsoEulerMaclaurin summation formula4.2 ReferencesAbel, N.H. (1823), Solution de quelques problmes laide dintgrales dniesButzer, P. L.; Ferreira, P. J. S. G.; Schmeisser, G.; Stens, R. L. (2011), The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signalanalysis, Results in Mathematics 59 (3):359400, doi:10.1007/s00025-010-0083-8, ISSN 1422-6383, MR2793463Olver, Frank W. J. (1997) [1974], Asymptotics and special functions, AKP Classics, Wellesley, MA: AKPetersLtd., ISBN 978-1-56881-069-0, MR 1429619Plana, G.A.A. (1820), Sur une nouvelle expression analytique des nombres Bernoulliens, propre exprimeren termes nis la formule gnrale pour la sommation des suites, Mem. Accad. Sci. Torino 25: 40341884.3. EXTERNAL LINKS 94.3 External linksAnderson, David, Abel-Plana Formula, MathWorld.Chapter 5BochnerRiesz meanThe BochnerRiesz mean is a summability method often used in harmonic analysis when considering convergenceof Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modication of the Riesz mean.Dene()+=_, if > 00, otherwise.Let f be a periodic function, thought of as being on the n-torus, Tn, and having Fourier coecientsf(k) for k Zn. Then the BochnerRiesz means of complex order , BRf of (where R > 0 and Re() > 0 ) are dened asBRf() =kZn|k|R_1 |k|2R2_+f(k)e2ik.Analogously, for a function f on Rnwith Fourier transformf() , the BochnerRiesz means of complex order ,SRf (where R > 0 and Re() > 0 ) are dened asSRf(x) =||R_1 ||2R2_+f()e2ixd.For >0 andn=1 ,SR andBR may be written as convolution operators, where the convolution kernel is anapproximate identity. As such, in these cases, considering the almost everywhere convergence of BochnerRieszmeans for functions inLpspaces is much simpler than the problem of regular almost everywhere convergenceof Fourier series/integrals (corresponding to =0 ). In higher dimensions, the convolution kernels become morebadly behaved (specically, for n12, the kernel is no longer integrable) and establishing almost everywhereconvergence becomes correspondingly more dicult.Another question is that of for which and which p the BochnerRiesz means of an Lpfunction converge in norm.This is of fundamental importance for n 2 , since regular spherical normconvergence (again corresponding to = 0) fails in Lpwhen p = 2 . This was shown in a paper of 1971 by Charles Feerman.[1] By a transference result, the Rnand Tnproblems are equivalent to one another, and as such, by an argument using the uniformboundedness principle,for any particular p (1, ) , Lpnorm convergence follows in both cases for exactly those where (1 ||2)+ isthe symbol of an Lpbounded Fourier multiplier operator. For n=2 , this question has been completely resolved,but for n 3 , it has only been partially answered. The case of n = 1 is not interesting here as convergence followsfor p (1, ) in the most dicult = 0 case as a consequence of the Lpboundedness of the Hilbert transform andan argument of Marcel Riesz.105.1. REFERENCES 115.1 References[1] Feerman, Charles (1971). The multiplier problemfor the ball. Annals of Mathematics 94 (2): 330336. doi:10.2307/1970864.5.2 Further readingLu, Shanzhen (2013). Bochner-Riesz Means on Euclidean Spaces (First ed.). World Scientic. ISBN 978-981-4458-76-4.Grafakos, Loukas (2008). Classical Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09431-1.Grafakos, Loukas (2009). Modern Fourier Analysis (Second ed.). Berlin: Springer. ISBN978-0-387-09433-5.Stein, Elias M. & Murphy, Timothy S. (1993). Harmonic Analysis: Real-variable Methods, Orthogonality, andOscillatory Integrals. Princeton: Princeton University Press. ISBN 0-691-03216-5.Chapter 6Borel summationBorel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classicaldivergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leer, who was the recognized lordof complex analysis. Mittag-Leer listened politely to what Borel had to say and then, placing his hand upon thecomplete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.Mark Kac, quoted by Reed & Simon (1978, p. 38)In mathematics, Borel summation is a summation method for divergent series, introduced by mile Borel (1899).It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum forsuch series. There are several variations of this method that are also called Borel summation, and a generalization ofit called Mittag-Leer summation.6.1 DenitionThere are (at least) three slightly dierent methods called Borel summation. They dier in which series they can sum,but are consistent, meaning that if two of the methods sum the same series they give the same answer.Throughout let A(z) denote a formal power seriesA(z) =k=0akzkand dene the Borel transform of A to be its equivalent exponential seriesBA(t) k=0akk! tk.6.1.1 Borels exponential summation methodLet An(z) denote the partial sumAn(z) =nk=0akzk.A weak form of Borels summation method denes the Borel sum of A to belimtetn=0tnn!An(z).126.2. BASIC PROPERTIES 13If this converges at z C to some a(z), we say that the weak Borel sum of A converges at z, and write akzk=a(z) (wB) .6.1.2 Borels integral summation methodSuppose that the Borel transform converges for all real numbers to a function growing suciently slowly that thefollowing integral is well dened (as an improper integral), the Borel sum of A is given by0etBA(tz) dt.If the integral converges at z C to some a(z), we say that the Borel sum of A converges at z, and write akzk=a(z) (B) .6.1.3 Borels integral summation method with analytic continuationThis is similar to Borels integral summation method, except that the Borel transform need not converge for all t, butconverges to an analytic function of t near 0 that can be analytically continued along the positive real axis.6.2 Basic properties6.2.1 RegularityThe methods (B) and (wB) are both regular summation methods, meaning that whenever A(z) converges (in thestandard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.k=0akzk= A(z) < akzk= A(z)(B,wB).Regularity of (B) is easily seen by a change in order of integration: if A(z) is convergent at z,thenA(z) =k=0akzk=k=0ak_0ettkdt_ zkk!=0etk=0ak(tz)kk!dt,where the rightmost expression is exactly the Borel sum at z.Regularity of (B) and (wB) imply that these methods provide analytic extensions to A(z).6.2.2 Equivalence of Borel and weak Borel summationAny series A(z) that is weak Borel summable at z C is also Borel summable at z. However, one can constructexamples of series which are divergent under weak Borel summation, but which are Borel summable. The followingtheorem characterises the equivalence of the two methods.Theorem ((Hardy 1992, 8.5)).Let A(z) be a formal power series, and x z C, then:1. If akzk= a(z) (wB) , then akzk= a(z) (B) .2. If akzk= a(z) (B) , and limt etBA(zt) = 0, then akzk= a(z) (wB) .14 CHAPTER 6. BOREL SUMMATION6.2.3 Relationship to other summation methods(B) is the special case of Mittag-Leer summation with = 1.(wB) can be seen as the limiting case of generalized Euler summation method (E,q) in the sense that as q the domain of convergence of the (E,q) method converges up to the domain of convergence for (B).[1]6.3 Uniqueness theoremsThere are always many dierent functions with any given asymptotic expansion. However there is sometimes a bestpossible function, in the sense that the errors in the nite-dimensional approximations are as small as possible in someregion. Watsons theorem and Carlemans theorem show that Borel summation produces such a best possible sum ofthe series.6.3.1 Watsons theoremWatsons theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that f is afunction satisfying the following conditions:f is holomorphic in some region |z| 1 to the function (s) and that (s) is analytic for (s) 1, except for a simple pole at s = 1 withresidue 1: that is,f(s) 1s 1is analytic in (s) 1. Then the limit as x goes to innity of exA(x) is equal to 1.31.2 ApplicationAn important number-theoretic application of the theorem is to Dirichlet series of the formn=1a(n)nswhere a(n) is non-negative. If the series converges to an analytic function in(s) bwith a simple pole of residue c at s = b, thennXa(n) cbXb.Applying this to the logarithmic derivative of the Riemann zeta function, where the coecients in the Dirichlet seriesare values of the von Mangoldt function, it is possible to deduce the PNT from the fact that the zeta function has nozeroes on the line8990 CHAPTER 31. WIENERIKEHARA THEOREM(s) = 1.31.3 ReferencesS. Ikehara (1931), An extension of Landaus theorem in the analytic theory of numbers, Journal of Mathe-matics and Physics of the Massachusetts Institute of Technology 10: 112, Zbl 0001.12902Wiener, Norbert (1932), Tauberian Theorems, Annals of Mathematics, Second Series 33 (1): 1100, doi:10.2307/1968102,ISSN 0003-486X, JFM 58.0226.02, JSTOR 1968102K. Chandrasekharan (1969). Introduction to Analytic Number Theory. Grundlehren der mathematischen Wis-senschaften. Springer-Verlag. ISBN 3-540-04141-9.Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridgetracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press. pp. 259266. ISBN 0-521-84903-9.Chapter 32Zeta function regularizationIn mathematics and theoretical physics, zetafunctionregularization is a type of regularization or summabilitymethod that assigns nite values to divergent sums or products, and in particular can be used to dene determinantsand traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but hasits origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.32.1 DenitionThere are several dierent summation methods called zeta function regularization for dening the sum of a possiblydivergent series a1 + a2 + ....One method is to dene its zeta regularized sum to be A(1) if this is dened, where the zeta function is dened forRe(s) large byA(s) =1as1+1as2+ if this sum converges, and by analytic continuation elsewhere.In the case when an = n, the zeta function is the ordinary Riemann zeta function, and this method was used by Eulerto sum the series 1 + 2 + 3 + 4 + ... to (1) = 1/12.Other values of s can also be used to assign values for the divergent sums (0)=1 + 1 + 1 + 1 + ... = 1/2, (2)=1 +4 + 9 + ... = 0 and in general (s) =n=1 ns= 1s+2s+3s+. . . = Bs+1s+1, where B is a Bernoulli number.[1]Hawking (1977) showed that in at space, in which the eigenvalues of Laplacians are known, the zeta functioncorresponding to the partition function can be computed explicitly. Consider a scalar eld contained in a large boxof volume V in at spacetime at the temperature T=1. The partition function is dened by a path integral overall elds on the Euclidean space obtained by putting =it which are zero on the walls of the box and which areperiodic in with period .In this situation from the partition function he computes energy, entropy and pressureof the radiation of the eld . In case of at spaces the eigenvalues appearing in the physical quantities are generallyknown, while in case of curved space they are not known: in this case asymptotic methods are needed.Another method denes the possibly divergent innite product a1a2.... to be exp(A(0)). Ray & Singer (1971)used this to dene the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifoldin their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of As.Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then theMinakshisundaramPleijel zeta function converges and has an analytic continuation as a meromorphic function to allcomplex numbers, and Seeley (1967) extended this to elliptic pseudo-dierential operators Aon compact Riemannianmanifolds. So for such operators one can dene the determinant using zeta function regularization. See "analytictorsion.Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta functionregularization in order to calculate the partition functions for thermal graviton and matters quanta in curved back-ground such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin9192 CHAPTER 32. ZETA FUNCTION REGULARIZATIONtransformation to the trace of the kernel of heat equations.32.2 ExampleThe rst example in which zeta function regularization is available appears in the Casimir eect, which is in a atspace with the bulk contributions of the quantum eld in three space dimensions. In this case we must calculate thevalue of Riemann zeta function at 3, which diverges explicitly. However, it can be analytically continued to s=3where hopefully there is no pole, thus giving a nite value to the expression. A detailed example of this regularizationat work is given in the article on the detail example of the Casimir eect, where the resulting sum is very explicitlythe Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive innity,leaving a physically signicant nite number).An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of aparticle eld in quantum eld theory. More generally, the zeta-function approach can be used to regularize the wholeenergy-momentum tensor in curved spacetime.The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modesof the vacuum:0|T00|0 =n|n|2Here, T00 is the zeroth component of the energy-momentum tensor and the sum (which may be an integral) isunderstood to extend over all (positive and negative) energy modesn ; the absolute value reminding us that theenergy is taken to be positive. This sum, as written, is usually innite ( n is typically linear in n). The sum may beregularized by writing it as0|T00(s)|0 =n|n|2|n|swhere s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensionalspace), the sum is manifestly nite, and thus may often be evaluated theoretically.The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physicalsystem are preserved. Zeta-function regularization is used in conformal eld theory, renormalization and in xingthe critical spacetime dimension of string theory.32.3 Relation to other regularizationsWe can ask if are there any relation to the dimensional regularization originated by the Feynman diagram. But nowwe may say they are equivalent each other. ( see .) However the main advantage of the zeta regularization is thatit can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside thecalculations i,j,k32.4 Relation to Dirichlet seriesZeta-function regularization gives a nice analytic structure to any sums over an arithmetic function f(n). Such sumsare known as Dirichlet series. The regularized formf(s) =n=1f(n)ns32.5. HEAT KERNEL REGULARIZATION 93converts divergences of the suminto simple poles on the complex s-plane. In numerical calculations, the zeta-functionregularization is inappropriate, as it is extremely slowto converge. For numerical purposes, a more rapidly convergingsum is the exponential regularization, given byF(t) =n=1f(n)etn.This is sometimes called the Z-transform of f, where z = exp(t). The analytic structure of the exponential andzeta-regularizations are related. By expanding the exponential sum as a Laurent seriesF(t) =aNtN+aN1tN1+ one nds that the zeta-series has the structuref(s) =aNs N+ .The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may beconverted to the other by making use of the integral representation of the Gamma function:(s + 1) =0xsexdxwhich lead to the identity(s + 1) f(s + 1) =0tsF(t) dtrelating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurentseries.32.5 Heat kernel regularizationThe sumf(s) =nanes|n|is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the n cansometimes be understood as eigenvalues of the heat kernel.In mathematics, such a sum is known as a generalizedDirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the LaplaceStieltjestransform, in thatf(s) =0estd(t)where (t) is a step function, with steps of an at t= |n| .A number of theorems for the convergence of such aseries exist. For example, by the Hardy-Littlewood Tauberian theorem, ifL = limsupnlog |nk=1 ak||n|then the series for f(s) converges in the half-plane (s) > L and is uniformly convergent on every compact subsetof the half-plane (s) > L . In almost all applications to physics, one has L = 094 CHAPTER 32. ZETA FUNCTION REGULARIZATION32.6 HistoryMuch of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zetafunction regularization methods was done by G.H. Hardy and J. E. Littlewood in 1916 and is based on the applicationof the CahenMellin integral. The eort was made in order to obtain values for various ill-dened, conditionallyconvergent sums appearing in number theory.In terms of application as the regulator in physical problems, before Hawking (1977), J. Stuart Dowker and RaymondCritchley in 1976 proposed a zeta-function regularization method for quantum physical problems. Emilio Elizaldeand others have also proposed a method based on the zeta regularization for the integrals axmsdx , here xsisa regulator and the divergent integral depends on the numbers (s m) in the limit s 0 see renormalization. Alsounlike other regularizations such as dimensional regularization and analytic regularization, zeta regularization has nocounterterms and gives only nite results.32.7 See alsoGenerating functionPerrons formulaRenormalization1 + 1 + 1 + 1 + 1 + 2 + 3 + 4 + Analytic torsionRamanujan summationMinakshisundaramPleijel zeta functionZeta function (operator)32.8 References^ TomM. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag New York.(See Chapter 8.)"^ A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, Analytic Aspects of Quantum Fields,World Scientic Publishing, 2003, ISBN 981-238-364-6^ G.H. Hardy and J.E. Littlewood, Contributions to the Theory of the Riemann Zeta-Function and the Theoryof the Distribution of Primes, Acta Mathematica, 41(1916) pp. 119196. (See, for example, theorem 2.12)Hawking, S. W. (1977), Zeta function regularization of path integrals in curved spacetime, Communicationsin Mathematical Physics 55 (2): 133148, Bibcode:1977CMaPh..55..133H, doi:10.1007/BF01626516, ISSN0010-3616, MR 0524257^ V. Moretti, Direct z-function approach and renormalization of one-loop stress tensor in curved spacetimes,Phys. Rev.D 56, 7797 (1997).Minakshisundaram, S.; Pleijel, . (1949), Some properties of the eigenfunctions of the Laplace-operator onRiemannian manifolds, Canadian Journal of Mathematics 1: 242256, doi:10.4153/CJM-1949-021-5, ISSN0008-414X, MR 0031145Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds., Advances in Math7: 145210, doi:10.1016/0001-8708(71)90045-4, MR 029538132.8. REFERENCES 95Garca Moreta, Jos Javier http://prespacetime.com/index.php/pst/article/view/498 The Application of ZetaRegularization Method to the Calculation of Certain Divergent Series and Integrals Rened Higgs, CMB fromPlanck, Departures in Logic, and GR Issues & Solutions vol 4 N 3 prespacetime journal http://prespacetime.com/index.php/pst/issue/view/41/showTocHazewinkel, Michiel, ed. (2001), Zeta-function method for regularization, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4Seeley, R. T. (1967), Complex powers of an elliptic operator, in Caldern, Alberto P., Singular Integrals(Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics 10, Providence,R.I.: Amer. Math. Soc., pp. 288307, ISBN 978-0-8218-1410-9, MR 0237943^ J.S. Dowker and R. Critchley, Eective Lagrangian and energy-momentum tensor in de Sitter space, Phys.Rev.D 13, 3224 (1976).[1] Tao, Terence (10 April 2010). The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variableanalytic continuation.96 CHAPTER 32. ZETA FUNCTION REGULARIZATION32.9 Text and image sources, contributors, and licenses32.9.1 Text Abels summationformula Source: https://en.wikipedia.org/wiki/Abel{}s_summation_formula?oldid=675918849 Contributors: MichaelHardy, Bearcat, Xezbeth, Mathbot, A. Pichler, JRSpriggs, CRGreathouse, PerryTachett, Addbot, Xqbot, Anne Bauval, Sawomir Biay,Slawekb, ZroBot, SporkBot, Frietjes, Solomon7968, Qsq and Anonymous: 4 AbelstheoremSource: https://en.wikipedia.org/wiki/Abel{}s_theorem?oldid=660332274Contributors: Michael Hardy,Dominus,Schneelocke, Bjcairns, Charles Matthews, Giftlite, CryptoDerk, Klemen Kocjancic, Geschichte, Danski14, Msh210, Pion, Ilario, RyanReich, Gimboid13, Grammarbot, R.e.b., Dbsanfte, CiaPan, Chobot, YurikBot, RobotE, Pred, RDBury, BeteNoir, Melchoir, Eskimbot,A. Pichler, Hottiger, CRGreathouse, Hanche, Thijs!bot, Hannes Eder, Ttwo, Safemariner, Jimothy 46, PMajer, Jobu0101, Kmhkmh,JackSchmidt, Nicolae Coman, Addbot, Zorrobot, KamikazeBot, Bdmy, Thehelpfulbot, Citation bot 1, Mathtyke, TuHan-Bot, AvicBot,ZroBot, L Kensington, Apc31, Brad7777, SiriusLH, Kodip, Darvii, Joseph2302 and Anonymous: 18 Abelian and tauberian theorems Source: https://en.wikipedia.org/wiki/Abelian_and_tauberian_theorems?oldid=667033241 Contrib-utors: AxelBoldt, Michael Hardy, Schneelocke, Charles Matthews, Dysprosia, Jitse Niesen, Robbot, Giftlite, Golbez, Rich Farmbrough,Msh210, Drbreznjev, R.e.b., Dbsanfte, Mathbot, Sodin, RussBot, Cartan, NawlinWiki, SmackBot, Melchoir, Polyade, Bluebot, Sillyrabbit, Mets501, CRGreathouse, Vanish2, David Eppstein, MystBot, Addbot, Uncia, Citation bot,Sawomir Biay, Citation bot 1,Kiefer.Wolfowitz, Trappist the monk, Brad7777, Saehry, Jochen Burghardt, Limit-theorem, Mark viking, EurasianBayes and Anony-mous: 6 AbelPlana formula Source: https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula?oldid=649441537 Contributors: MichaelHardy, Bearcat, Giftlite, R.e.b., Whoisjohngalt, A. Pichler, David Eppstein, Yobot, Trappist the monk, Wcherowi, Deltahedron andAnonymous: 2 BochnerRiesz mean Source: https://en.wikipedia.org/wiki/Bochner%E2%80%93Riesz_mean?oldid=581767526 Contributors: MichaelHardy, Bender235, GregorB, Yobot, Yyyyyumn, Sawomir Biay, Tcnuk, Helpful Pixie Bot and Anonymous: 2 Borel summationSource: https://en.wikipedia.org/wiki/Borel_summation?oldid=671033202 Contributors: Michael Hardy, Charles Matthews,Jitse Niesen, Jason Quinn, Sigfpe, Rgdboer, Crislax, Count Iblis, Linas, Rjwilmsi, R.e.b., Dbsanfte, Wavelength, Curpsbot-unicodify,SmackBot, Melchoir, Silly rabbit, Crittens, A. Pichler, CmdrObot, Bons, Myasuda, Karl-H, Headbomb, Pablodiazgutierrez, Daniele.tampieri,TXiKiBoT, Arjayay, Addbot, Ginosbot, Ozob, Luckas-bot, Yobot, Citation bot, VladimirReshetnikov, Pfm77, Slawekb, ZroBot, HelpfulPixie Bot, Odaniel1, Brad7777, Jianluk91, DG-on-WP and Anonymous: 19 Cauchy principal value Source: https://en.wikipedia.org/wiki/Cauchy_principal_value?oldid=674642917 Contributors: Michael Hardy,Charles Matthews, Giftlite, Jmnbpt, D6, ZeroOne, Billlion, Oleg Alexandrov, YurikBot, Gwaihir, Sbyrnes321, SmackBot, Njerseyguy,Domitori,FilipeS, Thijs!bot,Zylorian,Cardamon,Stevvers,Mir76,ANONYMOUS COWARD0xC0DE, Akulo,Daniele.tampieri,Rocketman116, Lechatjaune, Brews ohare, Matthieumarechal, ClculIntegral, Addbot, Yobot, Citation bot, ThibautLienart, Lucien-BOT, Lambda(T), Sawomir Biay, 777sms, Vincent Semeria, ZroBot, Helpful Pixie Bot, CitationCleanerBot, BattyBot, Lemnaminor,Clebor42 and Anonymous: 27 Darbouxs formula Source: https://en.wikipedia.org/wiki/Darboux{}s_formula?oldid=607144469 Contributors: Michael Hardy, Bearcat,Giftlite, Xezbeth, R.e.b., RussBot, JJL, SmackBot, Berland, A. Pichler, David Eppstein, Bearian, Yobot and Anonymous: 2 Dimensional regularizationSource: https://en.wikipedia.org/wiki/Dimensional_regularization?oldid=639331453 Contributors: MichaelHardy, Phys, Lumidek, David Schaich, Xezbeth, Flammifer, Fwb22, Ronark, Rjwilmsi, R.e.b., Chobot, Conscious, Zunaid, SmackBot,Siebren, Silly rabbit, TriTertButoxy, Headbomb, Shambolic Entity, Natsirtguy, Mild Bill Hiccup, Addbot, Niout, AnomieBOT, Xqbot,Omnipaedista, Erik9bot, Fcametti, Naviguessor, Suslindisambiguator, Bibcode Bot, AEIOU29979 and Anonymous: 17 DivergentseriesSource: https://en.wikipedia.org/wiki/Divergent_series?oldid=670866416Contributors: AxelBoldt, XJaM, MichaelHardy, Chinju, Charles Matthews, Jitse Niesen, Hyacinth, Gandalf61, MathMartin, Tobias Bergemann, Giftlite, Gene Ward Smith,Waltpohl, Meddlin' Pedant, Rgdboer, Crislax, Dfeldmann, Msh210, Mrholybrain, Linas, Sympleko, Tygar, Triddle, NatusRoma, Dan-wbartlett, Salix alba, R.e.b., LMSchmitt, Dppowell, DerHannes, Kier07, Attilios, Melchoir, Dan Hoey, MalafayaBot, Silly rabbit, Daqu,Mgiganteus1, Jim.belk, Dchudz, ChrisCork, Bons, Mudd1, Gregbard, Ntsimp, Arturocl, Mglg, Oerjan, Dawnseeker2000, Albmont,David Eppstein, Tercer, F3et, Policron, VolkovBot, LokiClock, AlleborgoBot, Qwfp, DumZiBoT, MystBot, Addbot, Nordisk varg, Zor-robot, Ettrig, Luckas-bot, Xqbot, Drilnoth, FrescoBot, Sawomir Biay, Kiefer.Wolfowitz, Asllearner, Ali Abbasi7, FoxBot, Numericana,ClueBot NG, Guy vandegrift, Trevayne08, Brad7777, BattyBot, Pawe Ziemian, Kanghuitari, Faizan, DilatoryRevolution, TCMemoire,A.bt(w) and Anonymous: 45 Euler summation Source: https://en.wikipedia.org/wiki/Euler_summation?oldid=668367596 Contributors: Michael Hardy, Stevenj, An-drewman327, Giftlite, Crislax, Linas, Gadget850, Melchoir, Chris the speller, A. Pichler, Bons, CBM, Myasuda, Gromgull, DavidEppstein, TXiKiBoT, Yugsdrawkcabeht, Addbot, Ginosbot, Luckas-bot, Xqbot, Ripchip Bot, Slawekb, RealzGirlz and Anonymous: 7 EulerMaclaurin formula Source: https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula?oldid=663051876 Contribu-tors: AxelBoldt, Miguel~enwiki, RTC, Michael Hardy, Stevenj, Charles Matthews, Jitse Niesen, McKay, Robbot, Fredrik, Sverdrup,Giftlite, MuDavid, Crislax, Eric Kvaalen, Linas, Graham87, BD2412, Godzatswing, FlaBot, Mathbot, Bgwhite, Zarel, Merosonox,Brian Tvedt, JJL, SmackBot, RDBury, Kurykh, AdamSmithee, Berland, Giganut, Antares784, Beetstra, A. Pichler, Thijs!bot, Unifey~enwiki,.anacondabot, Magioladitis, Vanish2, Email4mobile, David Eppstein, VectorBundle, LokiClock, Sapphic, Logan, SieBot, Reuqr, Brewsohare, Kiensvay, Kbdankbot, Cuaxdon, Haklo, Lightbot, Legobot, Luckas-bot, Yobot, Angry bee, Citation bot, Xqbot, DSisyphBot, Fres-coBot, Citation bot 1, Shawnarchy, Tal physdancer, Achim1999, Graroo, Ripchip Bot, WilliamADon, ClueBot NG, Hjilderda, BG19bot,IluvatarBot, Michael.a.cohen, Brirush, Tentinator, Comp.arch, Nigellwh, Someone not using his real name, Monkbot, BruceMathSoft-wareGuy, Jorge Guerra Pires and Anonymous: 55 HadamardregularizationSource: https://en.wikipedia.org/wiki/Hadamard_regularization?oldid=641973717Contributors: MichaelHardy, Andreas Kaufmann, R.e.b., Alaibot, Headbomb, Cardamon, Daniele.tampieri, Robertang, Sednivo~enwiki, RjwilmsiBot, Glay-hours and Roberthong HardyLittlewood tauberiantheoremSource: https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_tauberian_theorem?oldid=646854121 Contributors: Michael Hardy, Giftlite, R.e.b., Sodin, RDBury, Durova, Leyo, Uncia, Yobot, Sawomir Biay, Citation bot 1,Trappist the monk, Slawekb, Helpful Pixie Bot, K9re11, Bvictorri and Anonymous: 332.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 97 Hlder summation Source: https://en.wikipedia.org/wiki/H%C3%B6lder_summation?oldid=649088968 Contributors: Michael Hardy,Rjwilmsi, R.e.b., Yobot, AnomieBOT, K9re11 and Ethically Yours Lambert summationSource: https://en.wikipedia.org/wiki/Lambert_summation?oldid=618857703 Contributors: Michael Hardy, Cris-lax, Silly rabbit, A. Pichler, Vanish2, DOI bot, Yobot, Citation bot, Citation bot 1 and Monkbot LittlewoodsTauberiantheoremSource: https://en.wikipedia.org/wiki/Littlewood{}s_Tauberian_theorem?oldid=646636920Con-tributors: R.e.b., Josve05a and K9re11 Mittag-Leer summation Source: https://en.wikipedia.org/wiki/Mittag-Leffler_summation?oldid=671037802 Contributors: MichaelHardy, Rgdboer, R.e.b. and Deltahedron Nachbins theoremSource: https://en.wikipedia.org/wiki/Nachbin{}s_theorem?oldid=649088516 Contributors: Michael Hardy, CharlesMatthews, Giftlite, Linas, Ruud Koot, Sodin, JoshuaZ, Rschwieb, A. Pichler, CRGreathouse, Karl-H, Classicalecon, Yinweichen, Yobot,Sawomir Biay, Mkelly86, Brad7777, K9re11 and Anonymous: 10 PerronsformulaSource: https://en.wikipedia.org/wiki/Perron{}s_formula?oldid=637793705Contributors: XJaM, TakuyaMurata,Skysmith, Charles Matthews, Jitse Niesen, Giftlite, Bender235, EmilJ, Oleg Alexandrov, Linas, JosephSilverman, Twold, A. Pichler,Karl-H, RobHar, VolkovBot, CYCC, ChrisHodgesUK, Inys~enwiki, Addbot, ZroBot, Kamina, Helpful Pixie Bot, Deltahedron, K9re11and Anonymous: 6 Poissonsummationformula Source: https://en.wikipedia.org/wiki/Poisson_summation_formula?oldid=675459784 Contributors: MichaelHardy, SebastianHelm, Stevenj, Charles Matthews, Robbot, Robinh, Giftlite, Jacoplane, LeYaYa, Abdull, TheObtuseAngleOfDoom,Gadykozma, Ub3rm4th, Rbj, LutzL, Light current, User24, SmackBot, RDBury, Metacomet, Bob K, Summentier, JoshuaZ, Jim.belk,A. Pichler, CRGreathouse, HenningThielemann, Mct mht, WillowW, Skittleys, Headbomb, Second Quantization, Thenub314, DavidEppstein, Rosiestep, Addbot, Coolwangyx, Ozob, Yobot, Bdmy, Sawomir Biay, Citation bot 1, Tcnuk, RjwilmsiBot, Slawekb, ZroBot,Helpful Pixie Bot, Solomon7968, Wdlang, Brad7777, Saung Tadashi, Suhagja, Jens VF and Anonymous: 24 Ramanujansummation Source: https://en.wikipedia.org/wiki/Ramanujan_summation?oldid=661533646 Contributors: The Anome,Giftlite, Dratman, Histrion, Crislax, EmilJ, Wavelength, SmackBot, Melchoir, Silly rabbit, Berland, Sammy1339, Atulsvasu, CBM, Cy-debot, JamesBWatson, Albmont, David Eppstein, NorthAce, Addbot, Raulshc, LucienBOT, FoxBot, Ybab321, Brad7777, Permafrost46,Kasuga and Anonymous: 24 Regularization(physics) Source: https://en.wikipedia.org/wiki/Regularization_(physics)?oldid=674735506 Contributors: Michael Hardy,Ancheta Wis, Fropu, David Schaich, Jrme, Linas, Rjwilmsi, Chobot, KasugaHuang, SmackBot, Silly rabbit, Xxanthippe, Maliz,Racepacket, VolkovBot, Yartsa, Phe-bot, Eebster the Great, Addbot, Yobot, Niout, Jim1138, Ulric1313, Materialscientist, Citation bot,Xqbot, Omnipaedista, , Astiburg, Naviguessor, Maschen, Greggp42, Bibcode Bot, AvocatoBot, Mark viking and Anonymous: 14 Resummation Source: https://en.wikipedia.org/wiki/Resummation?oldid=654380815 Contributors: Phys, Lumidek, Count Iblis, Rjwilmsi,Conscious, Dialectric, SmackBot, Rrburke, Yobot, Bibcode Bot, AHusain314 and Anonymous: 3 Riesz meanSource: https://en.wikipedia.org/wiki/Riesz_mean?oldid=674665920 Contributors: Michael Hardy, Charles Matthews, Giftlite,Rich Farmbrough, Bender235, Linas, RussBot, SmackBot, Silly rabbit, Addbot, Uncia, Luckas-bot, Constructive editor, Citation bot 1,Onkekabonke, EmausBot, K9re11 and Anonymous: 1 Series acceleration Source: https://en.wikipedia.org/wiki/Series_acceleration?oldid=544640922 Contributors: Michael Hardy, Loisel,Rich Farmbrough, Pt, Count Iblis, Linas, Planetneutral, Reyk, Lambiam, A. Pichler, Hair Commodore, Addbot, Yobot, Aliotra, R. J.Mathar, Rezabot, Teika kazura, Deltahedron and Anonymous: 5 SilvermanToeplitz theorem Source: https://en.wikipedia.org/wiki/Silverman%E2%80%93Toeplitz_theorem?oldid=622269123 Con-tributors: Michael Hardy, Charles Matthews, Psychonaut, MathMartin, Giftlite, Mathbot, RussBot, BeteNoir, Silly rabbit, Rschwieb,Rksrathore, Yobot, Omnipaedista, Kiefer.Wolfowitz, Brad7777 and Anonymous: 4 Slowly varying function Source: https://en.wikipedia.org/wiki/Slowly_varying_function?oldid=673067139 Contributors: Michael Hardy,Jitse Niesen, Oleg Alexandrov, Sodin, Michael Slone, Nbarth, Alaibot, Thijs!bot, LachlanA, VolkovBot, JL-Bot, ClueBot, Addbot,Mzracz, Yobot, JWroblewski, Citation bot, Bdmy, Megade, Citation bot 1, ClueBot NG, Widr, Brad7777, Limit-theorem, Heatna-tion heatnation and Anonymous: 3 Summation by parts Source: https://en.wikipedia.org/wiki/Summation_by_parts?oldid=673987174 Contributors: Enchanter, MichaelHardy, Charles Matthews, Stan Lioubomoudrov, EmilJ, Oleg Alexandrov, Linas, Shreevatsa, Julien Tuerlinckx, Chobot, YurikBot,FF2010, Radagast83, A. Pichler, Myasuda, Alphachimpbot, JAnDbot, Magioladitis, David Eppstein, Bernardofpc, DavidGSterling,ChrisHodgesUK, MystBot, Addbot, Yobot, Calle, , Bdmy, Charvest, FrescoBot, Tcnuk, Oracleofottawa, Tbennert, Brad7777, Delta-hedron and Anonymous: 22 Wieners tauberiantheoremSource: https://en.wikipedia.org/wiki/Wiener{}s_tauberian_theorem?oldid=672106119 Contributors: Ax-elBoldt, Michael Hardy, Schneelocke, Charles Matthews, Giftlite, Oleg Alexandrov, LOL, Pol098, Rjwilmsi, R.e.b., Sodin, Deodar~enwiki,RDBury, WAREL, CRGreathouse, David Eppstein, Snowbot, Addbot, Uncia, Yobot, Kilom691, ZroBot, Rahul1990gupta, K9re11 andAnonymous: 7 WienerIkehara theoremSource: https://en.wikipedia.org/wiki/Wiener%E2%80%93Ikehara_theorem?oldid=631039519 Contributors:Michael Hardy, Charles Matthews, Giftlite, R.e.b., Sodin, Mon4, Headbomb, Vanish2, DavidCBryant, Snowbot, Uncia, Yobot, Citationbot, K9re11, Monkbot and Anonymous: 4 Zeta functionregularization Source: https://en.wikipedia.org/wiki/Zeta_function_regularization?oldid=670941509 Contributors: MichaelHardy, TakuyaMurata, Charles Matthews, GreatWhiteNortherner, Giftlite, Tom harrison, Dratman, Lumidek, Linas, Salix alba, R.e.b.,Incnis Mrsi, Silly rabbit, Nberger, QFT, CRGreathouse, Karl-H, Headbomb, David Eppstein, Leyo, Policron, Lartoven, Brews ohare,Versus22, Addbot, Luckas-bot, Ptbotgourou, Niout, Intractable, Xqbot, Tom.Reding, Dewritech, ZroBot, Chris81w, Crown Prince,Bibcode Bot, Trevayne08, 123957a, BattyBot, Enyokoyama, Reak spoughly, Mark viking, Ardehali and Anonymous: 3132.9.2 Images File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based o of Image:Ambox scales.svg Original artist: Dsmurat (talk contribs)98 CHAPTER 32. ZETA FUNCTION REGULARIZATION File:Fibonacci_spiral_34.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/93/Fibonacci_spiral_34.svg License: Publicdomain Contributors: self-drawn in Inkscape Original artist: User:Dicklyon File:Lebesgue_Icon.svgSource: https://upload.wikimedia.org/wikipedia/commons/c/c9/Lebesgue_Icon.svgLicense: Public domainContributors: w:Image:Lebesgue_Icon.svg Original artist: w:User:James pic File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License:Public domain Contributors:Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham) File:Wiki_letter_w.svg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg License: Cc-by-sa-3.0 Contributors:? Original artist: ?32.9.3 Content license Creative Commons Attribution-Share Alike 3.0