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A branch value is the image of a ramification point of a holomorphic map. Away from all branch values, a proper holomorphic map is an ordinary unramified covering.

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.

Set $F(z,w)=w^2-z^n+1$. Its complex gradient is $(-n{z}^{n-1},2w)$, never zero on $F=0$: simultaneous zero would give $z=w=0$, which is not on the curve. The complex implicit function theorem therefore gives a one-dimensional complex manifold, hence a Riemann surface. Where $w\ne 0$ use $z$ as coordinate, and where $z\ne 0$ use $w$; the transition maps and both projections are holomorphic.

For $\pi$, a ramification point of a holomorphic map has $w=0$ and $z=\zeta$ with ${\zeta }^n=1$. In the coordinate $w$, $z-\zeta =w^2/(n{\zeta }^{n-1})+O(w^4)$, giving ramification index $2$. Thus there are $n$ such points, and their branch points are the $n$th roots of unity.

For $\tau$, a ramification point of a holomorphic map has $z=0,w=\pm i$. In the coordinate $z$, $w-w_0=z^n/(2w_0)+O(z^{2n})$, giving ramification index $n$ at both points and branch points $\pm i$.

Write $n=2m$. At infinity set $u=1/z$, $v=w{u}^m$; then $v^2=1-u^n$. At $u=0$ there are two smooth points, $v=\pm 1$, and $u$ is a local coordinate. The extended $\pi$ has a simple pole at each, so neither is ramified. The degree-two Riemann-Hurwitz formula for the map to the Riemann sphere gives $2g-2=2(-2)+n$, henceThe list for $\tau$ above concerns the original affine Riemann surface; on the compactification its poles at the two added points have order $m$ and are additionally ramified if $m>1$.

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.