On a generalized Stieltjes trasform

Joshi, Jay

doi:10.2140/pjm.1964.14.969

Cited by 9 publications

(6 citation statements)

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“…and the corresponding inversion function E(x), which serves to invert the transform is defined by the equation This conventional convolution has been extended a certain class of generalized functions [Zemanian,7, p.229-246] and their inversion formula has been proved to be still valid when the limiting operation in that formula is understood as weak convergence in the space D' of Schwartz distributions.Setting s = ex and t = e-t in the conventional generalized Laplace transform We obtain F(ex) =~~;~ 1-: e(x-t)l1 1F1 (A, B; -e(x-t))j(e-t)e-tdt or ex F(ex) =~i;~ 1-: e(x-t)(/3+l)1F1 (A, B; -e(x-t))f(e-t)dt or J(x) =~~;~ 1-: e(x-t)(/3+l)1F1 (A, B; -e(x-t))j(t)dtwhere J(x) =ex F(ex) andj(t) = f(e-t).As in Joshi(5] and using some results from Hirschman and Widder(4] (p.66), the inversion operator E(D) is given by E(D){ ex F(ex)} = E(D){ J(x)} = lim (-lfna-{3+xe(n+{3)x Dne-(f3+ri+l)x Dne-ax D-ne(a+ri)x F( ex)where D1 = de"' •…”

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confidence: 91%

Some Theorems on a Generalized Laplace Transform of Generalized Functions

Gupta¹,

Mahato²

1994

Tamkang J. Math.

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In this paper we extend the generalized Laplace transform \[F(s)=\frac{\Gamma(\beta+\eta+1)}{\Gamma(\alpha+\beta+\eta+1)}\int_0^\infty (st)^\beta\ _1F_1(\beta+\eta+1, \alpha+\beta+\eta+1; -st)f(t) dt\] where $f(t)\in L(0,\infty)$, $\beta\ge 0$, $\eta > 0$; to a class of generalized functions. We will extend the above transform to a class of generalized functions as a special case of the convolution transform and prove an inversion formula for it.

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confidence: 91%

Some Theorems on a Generalized Laplace Transform of Generalized Functions

Gupta¹,

Mahato²

1994

Tamkang J. Math.

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“…The essential feature of finite-part integration is to give meaning to term by term integration that leads to divergent integrals. Consider the generalized Stieltjes transform [9,10,11] (1.1) F(ω) = a 0 f (x) (ω + x) ρ dx, Date: December 4, 2020.…”

Section: Introductionmentioning

confidence: 99%

Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

Villanueva,

Galapon

2020

Preprint

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Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E. A. Galapon, Proc. R. Soc. A 473, 20160567 (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform a 0 f (x)/(ω + x) ρ dx under the assumption that the extension of f (x) in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of f (x). Finite part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function which involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function 3 F 2 are derived.

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“…This procedure is first introduced in [1] and applied in [2] to obtain both an exact and asymptotic representations of the generalized Stieltjes transform of integral orders for small values of the real positive parameter ω. We will do the same to evaluate (1) and take the limit a → ∞ to obtain the corresponding result for the generalized Stieltjes transform of non-integer order [3,4,5,6,7]. The need for a separate treatment stems from the fact that the nature of the singularity of the complex valued function (ω + z) −λ depends on λ: when λ is an integer n, the integrand in (1) has a pole at z = −ω of order n; on the other hand, when λ is non-integer, the integrand has a branch point at z = −ω instead.…”

Section: Introductionmentioning

confidence: 99%

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

Tica

Galapon

2019

Journal of Mathematical Physics

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The paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform S λ [f ] = ∞ 0 f (x)(ω + x) −λ dx about ω = 0 where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x) −λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express S λ [f ] itself as a complex contour integral. This led to the recovery of the terms missed by naive term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f (x) has a zero at the origin of order m = 0, 1, . . . such that m − λ < 0 the correction terms missed out by naive term by term integration give the dominant contribution to S λ [f ] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naive term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

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On a generalized Stieltjes trasform

Cited by 9 publications

References 1 publication

Some Theorems on a Generalized Laplace Transform of Generalized Functions

Some Theorems on a Generalized Laplace Transform of Generalized Functions

Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

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