New methods for analytical calculation of elliptic integrals, applied in various physical problems

TL;DR

This paper reviews elliptic integrals' applications in physics and introduces a novel analytical method for calculating zero-order elliptic integrals, enhancing precision for theoretical and experimental use.

Contribution

It presents a new analytical approach combining elliptic function theory techniques to improve elliptic integral calculations.

Findings

New method improves calculation accuracy

Applicable to GPS, cosmology, black hole physics

Enhances theoretical and experimental analysis

Abstract

A short review will be made of elliptic integrals, widely applied in GPS (Global Positioning System) communications (accounting for General Relativity Theory-effects), cosmology, Black hole physics and celestial mechanics. Then a novel analytical method for calculation of zero-order elliptic integrals in the Legendre form will be presented, based on the combination of several methods from the theory of elliptic functions: 1. the recurrent system of equations for higher-order elliptic integrals in two different representations. 2. uniformization of four-dimensional algebraic equations by means of the Weierstrass elliptic function 3.a variable transformation, inversely (quadratically) proportional to a new variable. The developed method is a step forward towards constructing analytical methods, which can improve the precision of the calculation of elliptic integrals, necessary both for theoretical and experimental problems.