Generating Very-High-Precision Frobenius Series with Apriori Estimates of Coefficients

TL;DR

This paper presents a method to accurately predict the coefficients of Frobenius series solutions to differential equations using Legendre transformations of WKB approximations, enabling very-high-precision computations.

Contribution

It introduces a novel approach to estimate Frobenius series coefficients beforehand, improving the efficiency of high-precision solutions for differential equations.

Findings

Coefficients can be predicted with high accuracy before computation.

Legendre transformation of WKB solutions effectively estimates series behavior.

Method enhances the precision and speed of solving differential equations.

Abstract

The Frobenius method can be used to compute solutions of ordinary linear differential equations by generalized power series. Each series converges in a circle which at least extends to the nearest singular point; hence exponentially fast inside the circle. This makes this method well suited for very-high-precision solutions of such equations. It is useful for this purpose to have prior knowledge of the behaviour of the series. We show that the magnitude of its coefficients can be apriori predicted to surprisingly high accuracy, employing a Legendre transformation of the WKB approximated solutions of the equation.