Asymptotic solutions of the translation equation
The authors deal with asymptotic developments of solutions \(F(x,t)\) of the translation equation \(F(x,s+t)=F(F(x,s), t)\). They consider asymptotic formulas for \(F(x, u/x^m) \approx \sum^\infty_{n=1} f_n (u)x^n\) when \(x \to+0 (- 0)\) for fixed \(u\) and \(m \in \mathbb{N}\). Using these results they also consider asymptotic expansions \(F(x,t)=\sum^\infty_{n=1} \sum^{n-1}_{\nu=1} a_{n \nu} (x) \ln^\nu (t)t^{-n}\) for \(t \to \infty\). There is also a discussion about using Lie series for their purpose. Moreover several examples are given.
scientific article; zbMATH DE number 785162
scientific article