The exact calculation of the edge components of the angular Fock coefficients

TL;DR

This paper develops explicit analytic formulas for edge components of angular Fock coefficients, extending previous work to arbitrary k and simplifying complex subcomponents using hypergeometric functions and computational tools.

Contribution

It introduces methods to calculate edge components of angular Fock coefficients for any k and provides explicit formulas for previously missing subcomponents.

Findings

Derived explicit formulas for edge components with k ≤ 8.

Developed a single-series representation for a specific subcomponent.

Expressed complex subcomponents using hypergeometric and elementary functions.

Abstract

The present paper constitutes the development of our previous work devoted to calculations of the angular Fock coefficients, $\psi_{k,p}(\alpha,\theta)$. The explicit analytic representations for the edge components $\psi_{k,0}^{(0)}$ and $\psi_{k,0}^{(k)}$ with $k\leq8$ are derived. The methods developed enable such a calculation for arbitrary $k$. The single-series representation for subcomponent $\psi_{3,0}^{(2e)}$ missed in the previous paper is developed. It is shown how to express some of the complicated subcomponents through the hypergeometric and the elementary functions, as well. Using the specific operator of the Wolfram Mathematica, the simple explicit representations for some complex mathematical objects under consideration are obtained.