Speaker
Description
Let $f$ be an algebraic function and $f_0$ be its germ at some point $z_0\in\mathbb C$. Padé approximants are the best rational approximations of a given degree for the germ $f_0$. The convergence and the asymptotic behaviour of Padé approximants is described by Stahl theory. Nevertheless, sometimes this theory is used incorrectly in practice even in industrial applications. We give one example, where such mistake led to collapses in global electrical networks. We explain this mistake and explain how it was fixed by Trias in 2012 due to correct understanding of Stahl theory. Further we consider such generalizations of Padé polynomials as Hermite--Padé polynomials. Unfortunatelly, for these polynomials there is no analogue of Stahl theory for general $f$. But, sometimes Hermite--Padé polynomials are also used in practical applications without theoretical justification. Moreover, in some problems they are much more effective than usual Padé polynomials. We give such example from the molecular chemistry and justify it in the model case, when the function $f$ is 3-valued.
