Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches

Zozulya, V. V.

doi:10.1007/s10444-014-9399-3

Cited by 12 publications

(7 citation statements)

References 99 publications

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“…and thus the power series in (11) converges absolutely with infinite radius of convergence. Thus, P n (s) is always well-defined over the whole complex plane.…”

Section: Divergent Integral Of Monomialmentioning

confidence: 94%

“…In particular, the notion of renormalization is often used in modern physics for removing the divergent part of the integral of some physical quantity [9]. There are also other methods such as weakly singular and hypersingular integral regularization based on the theory of distributions [10,11]. The end goal of this paper is to extend the domain of certain divergent integrals to give a formal finite solution in the sense of analytic continuation.…”

Section: Introductionmentioning

confidence: 99%

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Analytic Continuation of Divergent Integrals

Aghili¹

2021

Preprint

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In this work the improper integral of monomial µ(s) = ∞ 1 x −s dx is considered as continuous analogy of infinite series in the context of the Riemann zeta function ζ(s) = ∞ n=1 n −s . Both the integral and the sum of monomial functions diverge to infinity for s ∈ C with Re(s) ≤ 1, while they become convergent otherwise. This paper presents analytic continuation of divergent integral of monomial over the entire complex plane with the exception of the pole at one, similar to analytic continuation of the zeta-function. It is shown that the improper integral can be equivalently written in term of Dirichlet series by making term-by-term integration of the monomial over successive integer intervals and using the Newton's generalization of the binomial theorem. This allows us to establish an elegant relationship between the µ-function and ζ-function and a functional equation that are exploited to assign meaningful solution to the divergent integral in the sense of analytic continuation. Subsequently, it is proved that the µ-function is holomorphic everywhere except for a simple pole at s = 1 and that the analytic continuation solution is consistent with the Abel-Plana formula.

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“…and thus the power series in (11) converges absolutely with infinite radius of convergence. Thus, P n (s) is always well-defined over the whole complex plane.…”

Section: Divergent Integral Of Monomialmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Analytic Continuation of Divergent Integrals

Aghili¹

2021

Preprint

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“…As hyper singular integrals of indefinite forms respectively − 1 + v 2 v and − 1 − v 2 v , the solutions in terms of the Hadamard finite parts ( HFP ) [66–72], read…”

Section: The Shape Pre-functions and Functions Green Operators Anmentioning

confidence: 99%

“…, the solutions in terms of the Hadamard finite parts (HFP) [66][67][68][69][70][71][72], read the E sector contribution to the shape pre-function integral equals unity from Equation (33a) as normalized by (À 4p=8p 2 ), which calls for a null H contribution, with or without an additional term; ii) the integral from Equation (33b) that characterizes the H sector contribution does have a null value, hence its product with the infinite (J ) factor is an undetermined form at that point; iii) the sum of the E and H parts of the shape function integral over the unit sphere will be normalized to unity if and only if (J )I T ø z (2) zÀHyp = 0 for the H term; iv) ensuring (J )I T ø z (2) zÀHyp = 0 needs to ''transform'' the infinite factor (J ) into a finite one; v) ''likely'' values to substitute with the À(4p=8p 2 )(8J ) factor a priori are (0, 61, 6i = ffiffiffiffiffiffi ffi À1 p ).…”

Section: The Normalization Operation On the Shape Pre-function Towardmentioning

confidence: 99%

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Franciosi

2020

Mathematics and Mechanics of Solids

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Since Eshelby’s (1957) result (Eshelby, JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc London A 1957; 421: 379–396) that ellipsoids in an infinite matrix have uniform localization tensors, all attempts to find other finite domain shapes sharing that same property have failed and valuable proofs were provided that none could exist. Since that “uniformity property” also applies to infinite cylinders and layers as limits of prolate and oblate spheroids, we examine the cases of hyperboloidal domains of which infinite cylinders and platelets also are the limits. As members of the quadric surface family, hyperboloids expectably also have uniform Eshelby tensor and Green operator when embedded in infinite media, with specific features expectable too from unboundedness and not convex curvatures. Using the Radon transform method applied by the author to various inclusion shapes, as well as to finite and infinite patterns, since the uniformity of a shape function (inverse Radon transform of the domain indicator function) implies the uniformity of the related Green operator and Eshelby tensor, we examine the shape functions of axially symmetric hyperboloids. We establish that those of the two-sheet types are uniform and those of the one-sheet types are not, an additional neck-related term carrying the non-uniformity. The Green operators are next examined in considering an isotropic embedding medium with elastic- (including dielectric-) like properties. The results regarding the operator (non) uniformity correspond to those concerning the shape functions. The established operator uniformity characteristics imply validity of all Eshelby-derived ellipsoid properties. Yet, determining the Green operator solution calls for overcoming the issue of the infinite hyperbolic planar sections (the operator finiteness), with also attention being paid to positive definiteness. Options are compared from which an obtained satisfying solution with regard to both issues raises questioning theoretical and practical points on mathematical and mechanical grounds. While further studies are in progress, some application tracks are indicated.

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“…However, it is the nature of divergent integrals that they take on a spectrum of values so that no unique value can be assigned to a divergent integral. The different values arise from distinct ways of interpreting divergent integrals corresponding to specific means of removing the divergences in the integration, such as by analytic continuation [4], by regularization [5], by summability methods [6], and by finite part integrals [7,8], among others [9,10,11,12,13]. The different values lead to different results for the target quantity to calculate and ambiguity inevitably arises as to which is the correct one [14,15].…”

Section: Introductionmentioning

confidence: 99%

Integration by divergent integrals: Calculus of divergent integrals in term by term integration

Galapon

2017

Preprint

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It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that merely assigning values to the divergent integrals leads to missing terms. This problem of missing terms was earlier resolved in the complex plane, which required the divergent integrals to assume values equal to their finite parts under a set of rules or calculus governing the use of the finite part integral [E. A. Galapon, Proc. Roy. Soc. A 473, 20160567 (2017)]. However, divergent integrals assume a spectrum of values arising from the different ways of removing the divergences in the integration. Here we show that the other interpretations of divergent integrals follow the same set of rules as those of the finite part integral, and then formulate the calculus of divergent integrals arising from term-by-term integration. We find that the calculus does not prescribe a specific interpretation to the divergent integrals but allows us to assign arbitrary values to them, with the missing terms automatically emerging from the chosen interpretation of the divergent integrals itself.

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Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches

Cited by 12 publications

References 99 publications

Analytic Continuation of Divergent Integrals

Analytic Continuation of Divergent Integrals

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Integration by divergent integrals: Calculus of divergent integrals in term by term integration

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