Asymptotics of the Humbert functions $\Psi_1$ and $\Psi_2$

TL;DR

This paper presents new asymptotic results for Humbert functions $$ and $$, confirms a conjecture related to the Glauber-Ising model, and introduces two elementary methods with broad applicability.

Contribution

It introduces two novel elementary asymptotic methods and applies them to Humbert functions and the Appell function, confirming a conjecture and expanding analytical tools.

Findings

Confirmed a conjectured limit in the Glauber-Ising model

Developed two elementary asymptotic methods

Demonstrated methods' effectiveness through examples

Abstract

A compilation of new results on the asymptotic behaviour of the Humbert functions $\Psi_1$ and $\Psi_2$, and also on the Appell function $F_2$, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the $1D$ Glauber-Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.