# Analytic image

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An extension of the concept of a complete analytic function, obtained on considering all possible elements of an analytic function in the form of generalized power series (Puiseux series) $$\begin{equation}\label{*} \sum_{\nu=m}^\infty a_\nu (z-z_0)^{\nu/n}\ , \ \ \ \sum_{\nu=m}^\infty a_\nu z^{-\nu/n} \end{equation}$$ Here $z$ is a complex variable, $m$ is an integer and $n$ is a natural number. The series converge in the domains $|z-z_0| < r$ and $|z| > r > 0$, respectively. An analytic image can be identified with the class of all elements of the form \eqref{*} which are obtained from each other by analytic continuation. The analytic image differs from the complete analytic function by the addition of all ramified elements of the form \eqref{*} with $n>1$, which are obtained by analytic continuation of its regular elements with $n=1$. After the introduction of a suitable topology, the analytic image is converted to the Riemann surface of the given function.