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

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 $gammain{mathbb R}setminus{0}$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of a such type formal solution of the third Painleve equation is presented and the proposed sufficient condition is applied to check its convergence.