The electrostatic interactions between charged particles immersed in an ionic solution play an important role for the stability of colloidal dispersions. The distribution of the electrostatic potential in the continuous phase obeys the nonlinear Poisson–Boltzmann equation. This study aims to calculate the asymptotic expressions for the interaction force and torque between two parallel charged c...

Compact expressions for spherically averaged position and momentum density integrals are given in terms of spherical Bessel functions (j(n)) and modified spherical Bessel functions (i(n)), respectively. All integrals required for ab initio calculations involving s, p, d, and f-type Gaussian functions are tabulated, highlighting a neat isomorphism between position and momentum space formulae. Sp...

We construct martingales whose 1-dimensional marginals are those of a centered selfdecomposable variable multiplied by some power of time t. Many examples involving quadratic functionals of Bessel processes are discussed.

Evaluation of integrals Meijer G-function Bessel functions Cosine and sine functions a b s t r a c t This paper is devoted to the study of the special function Z ν ρ (z) of z ∈ C with two parameters ρ ∈ R and ν ∈ C. In a special case such a function coincides with the McDonald function. Using the representations of Z ν ρ (z) in terms of the H-function, being different for ρ > 0 and ρ < 0, we de...

The object of this paper is to give new expressions for the wave field produced when a time harmonic point source is diffracted by a wedge with Dirichlet or Neumann boundary conditions on its faces. The representation of the total field is expressed in terms of quadratures of elementary functions, rather than Bessel functions, which is usual in the literature. An analogous expression is given f...

This paper provides novel analytic expressions for the incomplete Toronto function, TB(m,n, r), and the incomplete Lipschitz-Hankel Integrals of the modified Bessel function of the first kind, Iem,n(a, z). These expressions are expressed in closed-form and are valid for the case that n is an odd multiple of 1/2, i.e. n±0.5 ∈ N. Capitalizing on these, tight upper and lower bounds are subsequentl...

The suberized netting tissue on the surface of muskmelons is a remnant rind fragmentation caused by mechanical imbalance between expanding flesh meat and solidified epidermis during growth. appearance this pattern an important indicator for assessing quality muskmelons; however, very few quantitative studies have been conducted geometry netting. This study statistically analyzed areas fine frag...

A concise method of deriving expressions for Gaussian-like solutions the paraxial and wave equation (i.e. 3D d’Alembert equation) is presented. This based on Hankel transform. Choosing some Gaussian base functions with slight modifications prefactor all basic beams cylindrical character can be easily obtained. refers to Gaussian, Bessel-Gaussian, modified Laguerre–Gaussian Kummer-Gaussian (i.e....

We study in this paper the small data Cauchy problem for semilinear generalized Tricomi equations with a nonlinear term of derivative type $$\begin{aligned} u_{tt}-t^{2m}\Delta u=|u_t|^p \end{aligned}$$ $$m\ge 0$$ . Blow-up result and lifespan estimate from above are established $$1<p\le 1+\frac{2}{(m+1)(n-1)-m}$$ If $$m=0$$ , our results coincide those wave equation. The novelty consists const...