Related papers: An introduction to the Barnes double gamma functio…
We solve exactly the scalar box integral using the Mellin-Barnes representation. Firstly we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we perform an analytic continuation before…
Dyson's integration theorem is widely used in the computation of eigenvalue correlation functions in Random Matrix Theory. Here we focus on the variant of the theorem for determinants, relevant for the unitary ensembles with Dyson index…
We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental…
We consider the two-dimensional non-relativistic Coulomb problem with the aid of the momentum space construction of the associated Green's function. Our presentation has precursors in three dimensions. It is mainly Schwinger's approach…
We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. The theory of univalent functions plays a basic role in our work.
In the present paper we use the expansion formula of the polylogarithm function to construct approximations of the Caputo derivative which are related to the midpoint approximation of the integral in the definition of the Caputo derivative.…
We study translative integral formulas for certain translation invariant functionals on convex polytopes and discuss local extensions and applications to Poisson processes and Boolean models.
Mellin-Barnes (MB) representations have become a widely used tool for the evaluation of Feynman loop integrals appearing in perturbative calculations of quantum field theory. Some of the MB integrals may be solved analytically in closed…
In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give…
The multiple gamma functions of BM (Barnes-Milnor) type and the $q$-multiple gamma functions have been studied independently. In this paper, we introduce a new generalization of the multiple gamma functions called the $q$-BM multiple gamma…
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the…
The purpose of this paper is point out connections between scattering theory, double operator integrals, Kreins spectral shift function, integration theory, bimeasures, Feynman path integrals, harmonic and functional analysis and many other…
In the paper, we present a monotonicity result of a function involving the gamma function and the logarithmic function, refine a double inequality for the gamma function, and improve some known results for bounding the gamma function.
A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…
We introduce a n-term quadrature to integrate inner products of n functions, as opposed to a Gaussian quadrature to integrate 2n functions. We will characterize and provide computational tools to construct the inner product quadrature, and…
In this paper, certain generalized fractional derivative formulae are introduced involving the k-Mittag-Leffler function. Then their image formulae (using Beta transform, Laplace transform and Whittaker transform) are also established. The…
We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.…
A simple algorithm for the inverse scattering approach to the Camassa-Holm equation is presented.
We consider a model of 2D gravity with the action quadratic in curvature and represent path integrals as integrals over the SL(2, R) invariant Gaussian functional measure. We reduce these path integrals to the products of Wiener path…
The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals $\int_0^\infty x^{-2\nu}\cos(bx)\gamma(\nu,\alpha x^2)\,\mathrm{d}x$ (for $\nu>1/2$, $b>0$…