Analytic image

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.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)