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Let $x\in X$ and choose holomorphic coordinate charts centred at $x$ and $f(x)$. A nonconstant holomorphic map has a local expansion $w\circ f(z)=c{z}^m+O(z^{m+1})$, where $c\ne 0$ and $m\ge 1$. Taking a local holomorphic $m$th root of the nonvanishing factor and changing the source coordinate gives the exact local form $w=z^m$. The integeris the ramification index of a holomorphic map. A ramification point of a holomorphic map is a source point with ${\nu }_f(x)>1$, equivalently where its derivative vanishes; a branch value of a holomorphic map, also called a branch point here, is the image of such a point in $Y$. These are different sets, one in the source and one in the target.

Every fiber of a function is finite: locally it is the zero set of a nonzero holomorphic function, so its points are isolated, and compactness prevents infinitely many such zeros without an accumulation point. The same argument applied to the local derivatives makes the set of ramification points of a holomorphic map finite. The open mapping theorem and compactness show that $f(X)$ is both open and closed in the connected target $Y$, hence $f$ is surjective.

For any $y\in Y$, the degree of a holomorphic map isTo see independence of $y$, choose disjoint small coordinate charts about its finitely many preimages. The local form $z\mapsto z^{{\nu }_f(x)}$ shows that nearby values have ${\nu }_f(x)$ preimages counted with multiplicity in the corresponding chart. Compactness excludes additional preimages outside these charts for sufficiently nearby values: a sequence of such preimages would have a limit in $f^{-1}(y)$. The displayed sum is therefore locally constant, hence constant on connected $Y$. Away from branch values of a holomorphic map it is the ordinary number of preimages, and $f$ is a $d$-sheeted covering map there.

The Riemann-Hurwitz formula isor, in terms of the two genera,For its proof, choose a triangulation of a surface on $Y$ containing every branch value of a holomorphic map among its vertices. Every open edge and open face has $d$ lifts, since its interior avoids branch values of a holomorphic map. The lifted cells give a finite CW complex structure on $X$. If $V_Y,E_Y,F_Y$ are the target counts and $R={\sum }_x({\nu }_f(x)-1)$, the source counts areIndeed, at a target vertex the deficit from $d$ distinct preimages is precisely ${\sum }_{x\mapsto y}({\nu }_f(x)-1)$, and all nonzero contributions occur over vertices. Taking the alternating sum gives $\chi (X)=d(V_Y-E_Y+F_Y)-R$, as required. This is the Triangulation proof of the Riemann-Hurwitz formula.

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.

In local coordinates centred at $p$ and $f(p)$, a nonconstant rational map has the formThe point $p$ is a ramification point of a holomorphic map when its ramification index of a holomorphic map satisfies $e_p>1$. A branch value of a holomorphic map, also called a branch point in the target, is a value $f(p)$ of some ramification point.

Let $B$ be the finite set of branch values and put $R=f^{-1}(B)$. For every $w\notin B$, all points of $f^{-1}(w)$ have local degree one, so the holomorphic inverse function theorem supplies disjoint neighbourhoods on which $f$ is biholomorphic. Compactness of the fibre lets their target neighbourhoods be intersected to one evenly covered neighbourhood of $w$. Henceis an unramified covering map. Here the displayed restriction requires $R=f^{-1}(B)$; if “ramification point” is reserved only for points with $e_p>1$, then the source deletion must be written $f^{-1}(B)$.

The monodromy theorem says that analytic continuations along endpoint-fixed homotopic paths have the same terminal germ. Equivalently here, the path lifting theorem lifts a loop $\gamma$ based at $w\notin B$ from each point of $f^{-1}(w)$; taking the endpoint of each lift permutes that fibre. The permutation depends only on $[\gamma ]\in {\pi }_1({\mathbb{C}}_{\infty }\setminus B,w)$, giving the monodromy group of a covering.

For the stated function, make the Möbius change of coordinateThenThe critical points are $t=0,\pm 1/\sqrt{2},\infty$, with branch valuesA loop around $0$ interchanges the two roots born from $t=0$, so its branch cycle is a transposition. A loop around $-1/4$ simultaneously interchanges the two pairs that collide there, so its branch cycle is a product of two disjoint transpositions. With a suitable labelling these areThey generate a transitive group of order eight; their product is a four-cycle. Therefore the full monodromy group isthe dihedral group in its action on the four vertices of a square.

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

Let $f:X\to Y$ be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write $g_X,g_Y$ for their genera and $d\ge 1$ for the degree of a holomorphic map. With analytic ramification index $e_p$ at each $p\in X$, the formula isHere $\chi =2-2g$ is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives ${\sum }_{p\in f^{-1}(y)}{e}_p=d$ even at a branch value of a holomorphic map. Constant maps are excluded.^[ref]

The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of $d$ copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.

Count ramification points upstairs in $X$, weighted by $e_p-1$, rather than counting distinct branch values of a holomorphic map downstairs in $Y$. Grouping by fibres givesThe cardinality counts distinct points; the degree of a holomorphic map counts them with their analytic ramification indices. Thus each summand is the number of missing distinct preimages in that fibre.^[ref]

The phrase branch point varies across conventions: to keep the two surfaces distinct, use ramification point for $p\in X$ and branch value of a holomorphic map for $f(p)\in Y$. The existing name branch point of a holomorphic map here denotes the latter. Several ramification points can share one branch value of a holomorphic map.