Asymptotics of the Humbert functions ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$

A compilation of new results on the asymptotic behaviour of the Humbert functions ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$, 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.