On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem
| Autor: | Gontsov, Renat, Goryuchkina, Irina |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | "On the convergence of formal exotic series solutions of an ODE", Comput. Methods Funct. Theory, 2020, V. 20(2), 279-295 |
| Druh dokumentu: | Working Paper |
| Popis: | We propose a sufficient condition of the convergence of a complex power type formal series of the form $\varphi=\sum_{k=1}^{\infty}\alpha_k(x^{{\rm i}\gamma})\,x^k$, where $\alpha_k$ are functions meromorphic at the origin and $\gamma\in{\mathbb R}\setminus\{0\}$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of a such type formal solution of the third Painlev\'e equation is presented and the proposed sufficient condition is applied to check its convergence. Comment: 13 pages |
| Databáze: | arXiv |
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