Stieltjes Integral Representation Research Articles

Finite-part integration in the presence of competing singularities: Transformation equations for the hypergeometric functions arising from finite-part integration

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 ∫0af(x)/(ω+x)ρdx 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 that 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. In addition, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function 3F2 are derived.

Spectral analysis and computation for homogenization of advection diffusion processes in steady flows

Advective diffusion plays a key role in the transport of salt, heat, buoys, and markers in geophysical flows, in the dispersion of pollutants and trace gases in the atmosphere, and even in the dynamics of sea ice floes influenced by winds and ocean currents. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix D*. Three decades ago, a Stieltjes integral representation for the homogenized matrix, involving a spectral measure of a self-adjoint operator, was developed. However, analytical calculations of D* have been obtained for only a few simple flows, and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted. We overcome these limitations by providing a mathematical foundation for the computation of Stieltjes integral representations of D*. We explore two different approaches and for each approach we derive new Stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of D*, involving the molecular diffusivity and a spectral measure μ of a self-adjoint operator that depends on the characteristics of a randomly perturbed periodic flow. In discrete formulations of each approach, we express μ explicitly in terms of standard (or generalized) eigenvalues and eigenvectors of Hermitian matrices. We develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach. We use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties.

Internal resonances and relaxation memory kernels incomposites.

In heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

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

This paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform Sλ[f]=∫0∞f(x)(ω+x)−λdx 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 Sλ[f] itself as a complex contour integral. This led to the recovery of the terms missed by naïve 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 the naïve term by term integration give the dominant contribution to Sλ[f] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naïve 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.

Quantification of Lagrangian travel time statistics under nonstationary random groundwater flow conditions

AbstractThis study presents a Lagrangian description of advection dominated field‐scale transport of inert solutes, namely, the statistics of advective solute travel time and trajectory, in nonstationary random groundwater flow fields. The vertical flow field is generated by the effects of changes in total stress applied in fluid‐saturated heterogeneous deformable porous media. The Lagrangian statistics are developed directly from the velocity statistics based on the application of stochastic methodology such as the representation theorem and Fourier–Stieltjes integral representation. The focus of our numerical evaluation is placed on the influence of the soil compressibility on the Lagrangian statistics. The prediction on the statistics of solute travel time and trajectory may serve as the basic input to the environmental risk assessment.

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

The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f(z)(ω + z)−n. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient, and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

Spectral measure computations for composite materials

The analytic continuation method of homogenization theory provides Stieltjes integral representations for the eective parameters of composite media. These representations involve the spectral measures of self-adjoint random operators which depend only on the composite geometry. On nite bond lattices, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. Here we provide the mathematical foundation for rigorous computation of spectral measures for such composite media, and develop a numerically ecient projection method to enable such computations. This is accomplished by providing a unied formulation of the analytic continuation method which is equivalent to the original formulation and holds for nite and innite lattices, as well as in continuum settings. We also introduce a family of bond lattices and directly compute the associated spectral measures and eective parameters. The computed spectral measures are in excellent agreement with known theoretical results. The behavior of the associated eective parameters is consistent with the symmetries and theoretical predictions of models, and the computed values fall within rigorous bounds. Some previous calculations of spectral measures have relied on nding the boundary values of the imaginary part of the eective parameter in the complex plane. Our method instead relies on direct computation of the eigenvalues and eigenvectors which enables, for example, statistical analysis of the spectral data.

Stochastic analysis of stream–groundwater interaction subject to temporally correlated recharge

summary In this article, the coupled equations describing the transient groundwater flow and stream flow fluctuations are solved based on the Fourier–Stieltjes integral representation, where the lateral inflow term in the stream flow perturbation equation connects the process of groundwater flow to that of stream flow. The closed-form solutions describing the variability in groundwater flow specific discharge and stream flow discharge are developed to quantify the influence of the variability in temporally correlated recharge in an unconfined aquifer on the stream flow discharge. We found from the closed-form solutions that the temporal correlation scale and the variance of recharge fields influence the variability in groundwater flow specific discharge and the stream flow discharge positively. In addition, the hydraulic diffusivity coefficient takes the role of reducing the variability in stream flow discharge.

Stieltjes and other integral representations for functions of Lambert W

We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci and Sokal. In addition, we consider the relationship of functions of W with the class of completely monotonic functions and show that W is a complete Bernstein function.

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.

Characterization of structure and properties of bone by spectral measure method

Novel mathematical method called spectral measure method (SMM) is developed for characterization of bone structure and indirect estimation of bone properties. The spectral measure method is based on an inverse homogenization technique which allows to derive information about the structure of composite material from measured effective electric or viscoelastic properties. The mechanical properties and ability to withstand fracture depend on the structural organization of bone as a hierarchical composite. Information about the bone structural parameters is contained in the spectral measure in the Stieltjes integral representation of the effective properties. The method is based on constructing the spectral measure either by calculating it directly from micro-CT images or using measurements of electric or viscoelastic properties over a frequency range. In the present paper, we generalize the Stieltjes representation to the viscoelastic case and show how bone microstructure, in particular, bone volume or porosity, can be characterized by the spectral function calculated using measurements of complex permittivity or viscoelastic modulus. For validation purposes, we numerically simulated measured data using micro-CT images of cancellous bone. Recovered values of bone porosity are in excellent agreement with true porosity estimated from the micro-CT images. We also discuss another application of this method, which allows to estimate properties difficult to measure directly. The spectral measure method based on the derived Stieltjes representation for viscoelastic composites, has a potential for non-invasive characterization of bone structure using electric or mechanical measurements. The method is applicable to sea ice, porous rock, and other composite materials.

Asymptotic analysis of the Green-Kubo formula

[Received on 30 November 2009; revised on 10 May 2010; accepted on 13 May 2010] A detailed study of various distinguished limits of the Green–Kubo formula for the self-diffusion coefficient is presented in this paper. First, an alternative representation of the Green–Kubo formula in terms of the solution of a Poisson equation is derived when the microscopic dynamics is Markovian. Then the techniques developed in Golden & Papanicolaou (1983, Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys. , 90, 473–491) and Avellaneda & Majda (1991, An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows. Commun. Math. Phys. , 138, 339–391) are used to obtain a Stieltjes integral representation formula for the symmetric and antisymmetric parts of the diffusion tensor. The effect of irreversible microscopic dynamics on the diffusion coefficient is analysed and various asymptotic limits of physical interest are studied. Several examples are presented that confirm the findings of our theory.

Inverse Bounds of Two-Component Composites

A method is presented for estimating microstructural parameters from permittivity measurements of two-component composites. This structural information is described by a particular positive measure in the Stieltjes integral representation of the effective permittivity. The dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure. We present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments. These inverse bounds are improved when the volume fraction is known or the material is isotropic. Composites with known geometrical structure illustrate the method.

Nonlinear approximations to the derivative of delta function

The nonlinear approximations based on trigonometric generating functions are studied. It is shown that, by properly choosing generating functions, such nonlinear approximations to the derivative of the Dirac delta function on [−1, 1] are the corresponding Gaussian quadratures applied to its Stieltjes integral representations by the generating functions. Moreover, the approximations are proved to be convergent and the error terms are obtained.

Stieltjes integral representation of effective diffusivities in time-dependent flows

A Stieltjes integral representation for the effective diffusivity of a passive scalar in time-dependent, incompressible flows is developed. The representation provides a summability formula for the perturbative expansion of the diffusivity in powers of the P\'eclet number. In particular, upper and lower bounds on the effective diffusivity are obtained from Pad\'e approximants of the series.

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Peclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Peclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek0(Pe)1/2 in the limit of large Peclet numbers wherek0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

Stieltjes integral representation and effective diffusivity bounds for turbulent transport.

A Stieltjes integral representation for the effective diffusivity in turbulent transport is developed. This formula is valid for all Peclet numbers and yields a rigorous resummation of the divergent perturbation series in Peclet number provided that all diagrams are computed exactly. Another consequence of the integral representation is convergent upper and lower bounds on effective diffusivity for all Peclet numbers utilizing a prescribed finite number of terms in their perturbation series.

The Generating Function Method of Nonlinear Approximation

The generating function method is used to derive nonlinear approximations to a given function using any of a large variety of agreement conditions between function and approximation. The linear and fully nonlinear approximations coincide with Baker–Gammel approximants. While the latter exploits the correspondence between Padé approximants and Taylor series, the generating function method applies Newton–Cotes quadrature to the Stieltjes integral representation of the given function obtaining error terms from the quadrature errors.

The Gauss Hypergeometric Ratio As a Positive Real Function

The Gauss continued fraction for the ratio of two hypergeometric functions is converted into an ordinary fraction (all partial numerators are 1) and simplifications occurring for particular relations between the parameters are discussed. In particular, a very simple expansion is obtained for the ratio ${E /K}$ of the complete elliptic integrals. For the argument $ - z$ and for certain ranges of the parameters, the Gauss expansion is a Stieltjes fraction and represents the input impedance of a passive ladder network. The Stieltjes integral representations of the corresponding positive real functions are established and yield many new definite integrals. A general method for obtaining indefinite integrals involving independent solutions of a self-adjoins differential equation, by means of the Wronskian, is also mentioned. Finally, some continued fractions originating from other contiguity relations for hypergeometric functions are discussed.

Boundedness Properties of Stationary Linear Operators

A characterization of boundedness given by R. E. Lane provides a foundation for studying stationary, linear transformations which map the left-continuous quasi-continuous functions on the line into functions on the line. Properties of these operators are identified in terms of the transforms of $J(t) = \chi _{(0,\infty )} $ It is shown that the norm of an operator T is the variation of $TJ$, that there is a Stieltjes integral representation of T with this function as the integrator, and that convergence of a sequence of these operators is equivalent to convergence in variation of the sequence of transforms of J. An operator which is continuous under pointwise convergence is shown to be bounded on a closed interval, and is characterized by the left-continuity of $TJ$.