# Absolutely Convergent Series

The Experts below are selected from a list of 279 Experts worldwide ranked by ideXlab platform

### Alexey Kuznetsov – 1st expert on this subject based on the ideXlab platform

• ##### On the density of the supremum of a stable process
arXiv: Probability, 2011
Co-Authors: Alexey Kuznetsov

Abstract:

We study the density of the supremum of a strictly stable L\’evy process. As was proved recently in F. Hubalek and A. Kuznetsov “A Convergent Series representation for the density of the supremum of a stable process” (Elect. Comm. in Probab., 16, 84-95, 2011), for almost all irrational values of the stability parameter $\alpha$ this density can be represented by an Absolutely Convergent Series. We show that this result is not valid for all irrational values of $\alpha$: we construct a dense uncountable set of irrational numbers $\alpha$ for which the Series does not converge Absolutely. Our second goal is to investigate in more detail the important case when $\alpha$ is rational. We derive an explicit formula for the Mellin transform of the supremum, which is given in terms of Gamma function and dilogarithm. In order to illustrate the usefulness of these results we perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics, such as q-Series, Diophantine approximations and quantum dilogarithms, and we also suggest several open problems.

• ##### a ConvergentSeries representation for the density of the supremum of a stable process
Electronic Communications in Probability, 2011
Co-Authors: Friedrich Hubalek, Alexey Kuznetsov

Abstract:

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ – except for a dense set of Lebesgue measure zero – the asymptotic Series which were obtained in Kuznetsov (2010) “On extrema of stable processes” are in fact Absolutely Convergent Series representations for the density of the supremum.

### Bin‐hao Jiang – 2nd expert on this subject based on the ideXlab platform

• ##### Expansion of Spherical Wave Functions for Electro-Magnetic Fields of a Vertical Magnetic Dipole over an Anisotropic Half-Space
Chinese Journal of Geophysics, 2020
Co-Authors: Bin‐hao Jiang

Abstract:

We derive an accurate and analytical expression for electromagnetic fields of a vertical magnetic dipole over an uniaxial medium half-space. By using the expansion of spherical wave functions for a cylindrical wave function and with the help of the theory of hypergeometric functions, the Sommerfeld type integral contained in the electromagnetic fields is expressed as a rapidly and Absolutely Convergent Series of spherical wave functions; and the coefficients of the Series are the Legendre Polynomials with the complex argument for the constitutive parameter. The present results have explicit mathematical and physical interpretations and no the restriction of the locations of dipole source and observation points, medium parameters and frequency. The expression can be used conveniently to calculate and analyze electromagnetic fields at any points.

### Liu Yongtan – 3rd expert on this subject based on the ideXlab platform

• ##### Expansion of Spherical Wave Functions for Hertz Potential of a Vertical Magnetic Dipole over Conducting Medium Plane
, 2020
Co-Authors: Liu Yongtan

Abstract:

An accurate and efficient method for computing Sommerfeld type integrals is crucial in the analysis of the electromagnetic waves for radiators and scatters in the presence of a conducting half-space.In this paper,an analytical approach is described for the evaluation of the Hertz potential of a vertical magnetic dipole over the finitely conducting medium plane.By using the expansion of spherical wave functions for a cylindrical wave function and the techniques of the hypergeometric functions,the Sommerfeld integral contained in the potential is expressed as a rapidly and Absolutely Convergent Series of spherical wave functions;and the coefficients of the Series are cast into the Legender polynomials with complex argument for the medium parameter.The obtained result has the advantage that it holds without the restrictions on the locations of dipole source and observation points,ranges of medium parameters and frequency and so on,and can conveniently be employed to calculate and analyze the electromagnetic fields.