A unified approach to the integrals of Mellin–Barnes–Hecke type

In this paper, we provide a unified approach to a family of integrals of Mellin–Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by Hörmander. We derive by this method the integrals of Hecke and Sonine related to various types of Bessel functions which have found applications in analytic and algebraic number theory.     

On a generalization of Chen's iterated integrals

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In this paper, Chen's iterated integrals are generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive iterative property and comultiplication formula. In a particular example, a (non-classical) multiplicative iterative property is also shown to hold. After developing this theory in the first part of the paper we discuss various applications, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the existence of a contour integration proof of the functional equation for the Dedekind zeta function emerges); a way of thinking about complex iterated derivatives arising from a reformulation of a result of Gel'fand and Shilov in the theory of distributions; and a direct topological proof of the monodromy of polylogarithms.

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For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=dsVvo7s8BYU.     

Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Fractional integrals and derivatives of Airy functions (Riesz potentials) are considered. For half integrals D ^−1/2 Ai(x) and D ^−1/2 Gi(x) explicit representations are found in terms of the products of Airy functions. Here Ai(x) and Gi(x) are the Airy function of the first kind and the Scorer function, respectively. Based on that representations are obtained for all semi-integer derivatives of Ai(x) and Gi(x). Applications to Korteweg–de Vries type equations are provided.      

Numerical solution of some partial differential equations by means of a deterministic method of approximate functional integration

A numerical method of solution of some partial differential equations is presented. The method is based on representation of Green functions of the equations in the form of functional integrals and subsequent approximate calculation of the integrals with the help of a deterministic approach. In this case the solution of the equations is reduced to evaluation of usual (Riemann) integrals of relatively low multiplicity. A procedure allowing one to increase accuracy of the solutions is suggested. The features of the method are investigated on examples of numerical solution of the Schrödinger equation and related diffusion equation.     

The Bloch-Grüneisen integrals. Some analytic expressions

For the Bloch-Grüneisen integrals and some of their modifications we give some expressions which are convenient in analytic transformations or permit the numerical calculation of values of these integrals. This revised version was published online in June 2006 with corrections to the Cover Date.     

The superconvergence of the Newton-Cotes rule for Cauchy principal value integrals

We consider the general (composite) Newton-Cotes method for the computation of Cauchy principal value integrals and focus on its pointwise superconvergence phenomenon, which means that the rate of convergence of the Newton-Cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point. The necessary and sufficient conditions satisfied by the superconvergence point are given. Moreover, the superconvergence estimate is obtained and the properties of the superconvergence points are investigated. Finally, some numerical examples are provided to validate the theoretical results.     

On sampling theory and basic Sturm–Liouville systems

We investigate the sampling theory associated with basic Sturm–Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm–Liouville systems. Examples involving basic sine and cosine transforms are given.     

Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.     

The role of the Fox–Wright functions in fractional sub-diffusion of distributed order

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin–Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox–Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

An efficient algorithm for the Hurwitz zeta and related functions

A simple class of algorithms for the efficient computation of the Hurwitz zeta and related special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical examples are briefly described. The algorithms are easy to implement and compete with Euler–Maclaurin summation-based methods.     

Hypersingular kernel integration in 3D Galerkin boundary element method

We consider Cauchy singular and Hypersingular boundary integral equations associated with 3D potential problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. In particular, for constant and linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the Galerkin matrix elements. These numerical indications are given after an analysis of the singularities arising in the whole integration process, which is valid also for shape functions of higher degrees.     

Forward–backward systems for expected utility maximization

In this paper we deal with the utility maximization problem with general utility functions including power utility with liability. We derive a new approach in which we reduce the resulting control problem to the study of a system of fully-coupled Forward–Backward Stochastic Differential Equations (FBSDEs) that promise to be accessible to numerical treatment.     

Comment on a paper of Rao et al., an entry of Ramanujan and a new 3F2(1)

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].     

Extremal values of multiple gamma and sine functions

We study the extremal values of multiple gamma and sine functions in the fundamental intervals. We show the number and locations of the extremal points, and prove that all the local maximum and minimum values are greater and less than one, respectively.     

Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.