On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem

Renat Gontsov; Irina Goryuchkina

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arXiv · 1906.06716

On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem

Renat Gontsov·Irina Goryuchkina

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Abstract

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.

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BibTeX RIS

Renat Gontsov, Irina Goryuchkina. 2019-06-16. On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem. https://arxiv.org/abs/1906.06716

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