Fadhel Al-Musallam

Further results on a generalized gamma function occurring in diffraction theory

Recently, Kobayashi (Jour. Phy. Soc. Japan 60(1991), 1501-1512) has considered a generalized Gamma function, Г m (u,υ), occurring in Diffraction Theory. This article considers a more general function involving Gauss hypergeometric function F(a,b;c,z), that reduces to the generalized Gamma function when p = 1 and b = c. Several properties for the function D; including analytic continuation, recurrence formulae and computation for special values of the parameters, are given. Moreover, some new properties of the function Г m (u,υ) are established.

Asymptotic expansions for generalized gamma and incomplete gamma functions

Kobayashi(jour.phy.soc.japan 60(1991),1501_1512) considered a generalized Gamma function occuring in Diffraction Theory. This article considers a more general function involving Gauss hypergeometric function F(a,b;c,x) Where that reduces to when p=1 and b=c. The incomplete and the complementary incomplete functions associated with are introduced. Some properties and recurrence relations satisfied by these two functions are proved, and asymptotic expansions for the function and its associated incomplete and complementary incomplete functions are fully investigated.

Integral Transforms Related to a Generalized Convolution

AbstractIntegral transformations of the form from Lp(ℝ+) into Lq(ℝ+); (1 ≤ p ≤ 2, p−1+ q−1 = 1) are studied. Necessary and sufficient conditions on k to ensure that the transformation is unitary on L2(ℝ+) are obtained, and a formula for the inverse transformation is derived in this case.

Strong convergence of a hybrid steepest descent method for the split common fixed point problem

We study the convergence properties of an iterative method for a variational inequality defined on a solution set of the split common fixed point problem. The method involves Landweber-type operators related to the problem as well as their extrapolations in an almost cyclic way. The evaluation of these extrapolations does not require prior knowledge of the matrix norm. We prove the strong convergence under the assumption that the operators employed in the method are approximately shrinking.

Lagrange's formula for tangential interpolation with application to structured matrices

Lagranges interpolation formula is generalized to tangential interpolation. This includes interpolation by vector polynomials and by rational vector functions with prescribed pole characteristics. The formula is applied to obtain representations of the inverses of Cauchy-Vandermonde matrices generalizing former results.

Hermite's formula for vector polynomial interpolation with applications to structured matrices

The classical Hermite formula for polynomial interpolation is generalized to interpolation of vector polynomials (tangential interpolation). The formula exhibits a relation between the matricial homogeneous problem and the nonhomogeneous vector problem. As applications, a formula for the inverse of a generalized Vandermonde matrix is presented and it is shown that the well-known formulas for the inverse of a Toeplitz matrix can be obtained as a special case of the generalized Hermite formula.

A grid generator for problems on unbounded intervals with rationally decaying solutions

In this paper a grid generator adapted to rationally decaying solutions on unbounded intervals is proposed and analyzed. The approach is illustrated with specific examples and numerical results are given.ZusammenfassungIn der vorliegenden Arbeit wird ein Gittergenerator für Probleme auf unbeschränkten Intervallen mit rational abklingendem Lösungsverhalten vorgeschlagen und analysiert. Der gewählte Zugang wird anhand spezieller Beispiele illustriert, und es werden numerische Ergebnisse hierzu angegeben.

The equivalent mass density for the elastic scattering by many small rigid bodies and applications

We deal with the elastic scattering by a large number M of rigid bodies of arbitrary shapes with maximum radius a; 0 < a << 1 with constant Lam e coecients and . We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region of R 3 where their number is M := M(a) := O(a 1 ) and the minimum distance between them is d := d(a) a t with t in some appropriate range, as a! 0, the generated far-eld patterns approximate the far-eld patterns generated by an equivalent (possibly variable) mass density. This mass density is described by two coecients: one modeling the local distribution of the small bodies and the other one by their geometries. In particular, if the distributed bodies have a uniform spherical shape then the equivalent mass density is isotropic while for general shapes it might be anisotropic. In addition, we can distribute the small bodies in such a way that the equivalent mass density is negative. Finally, if the background density is variable in and = 1 in R 3 n , then if we remove from appropriately distributed small bodies then the equivalent density will be equal to unity in R 3 , i.e. the obstacle characterized by is approximately cloaked.

A Modified and a Finite Index Weber Transforms

This paper introduces, by way of constructing, specific finite and infinite integral transforms with Bessel functions Jν and Yν in their kernels. The infinite transform and its reciprocal look deceptively similar to the known Weber transform and its reciprocal, respectively, but fundamentally differ from them. The new transform enjoys an operational property that makes it useful for applications to some problems in differential equations with non-constant coefficients. The paper gives a characterization of the image of some spaces of square integrable functions with respect to some measure under the infinite and finite transforms.

H-FUNCTION WITH COMPLEX PARAMETERS I: EXISTENCE

An H-function with complex parameters is defined by a Mellin-Barnes type inte- gral. Necessary and sufficient conditions under which the integral defining the H-function converges absolutely are established. Some properties, special cases, and an application to integral transforms are given.