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A function element is a pair $(f,D)$ consisting of a domain $D\subseteq G$ and a holomorphic function $f:D\to \mathbb{C}$ belonging to the given complete analytic function. Its germ of a holomorphic function at $z\in D$ is the equivalence class $[f{]}_z$, where two elements are equivalent when they agree on a neighbourhood of $z$.

For every function element $(f,D)$ and sufficiently small open $U\subseteq D$, setThese sets form a basis for the space of germs of holomorphic functions. The projectionrestricts to a homeomorphism $U_f\to U$. The inverse projections $\pi {|}_{U_f}$ are compatible complex charts, making $\mathcal{G}$ a Riemann surface and $\pi$ a covering map and local biholomorphism. In such a chart, the evaluation map on a space of germs has coordinate expressionIt is therefore holomorphic on every chart, hence analytic on each component of $\mathcal{G}$.

Now let $f:R\to S$ be a nonconstant analytic map of compact connected Riemann surfaces, and let $B$ be its finite branch-value set. Each point of $B$ is a branch value of a holomorphic map. At every point of $R\setminus f^{-1}(B)$ the local degree is one, so the holomorphic inverse-function theorem supplies a neighbourhood on which $f$ is biholomorphic. For $y\in S\setminus B$, the fibre is finite. Choose disjoint inverse neighbourhoods around all its points and shrink their common image; compactness, or equivalently properness of $f$, ensures that no additional preimages enter. Thusis an ordinary unramified covering map.

Fix $P\in S\setminus B$. Given a loop $\gamma$ based at $P$, lift it from every $Q\in f^{-1}(P)$ and send $Q$ to the endpoint of its lift. The path lifting theorem makes this a permutation; concatenation of loops composes the permutations and reversal gives inverses. Its image is the monodromy group of a covering. The punctured surface $R\setminus f^{-1}(B)$ is path connected. Given two points of the fibre, join them there by a path; its projection is a loop whose lift joins those points. Hence the monodromy action is transitive.

Finally considerIt has degree four. Its finite critical points satisfywith branch values $2$ and $-2$; the double poles at zero and infinity give the branch value infinity. For a regular value, the fibre isThe two commuting involutions $z\mapsto -z$ and $z\mapsto z^{-1}$ preserve $F$ and act transitively on this fibre. They generate all four deck transformations, so the covering is normal and its monodromy group isWith the displayed ordering of the fibre, its elements areThis is the monodromy group of z squared plus z to the minus two.