Harmonic conjugates and integral logarithmic-pole monodromy on surfaces

Local conjugates exist on small chart discs. Let $W\subseteq X$ be open and $u$ harmonic on $W$. For every $x\in W$ choose a chart $\phi :U\to \mathbb{C}$ of $X$ with $x\in U$; then $u_{\phi }$ is harmonic on the open set $\phi (U\cap W)$ by [F1]. By [F2] there is a radius $r>0$ with $D(\phi (x),r)\subseteq \phi (U\cap W)$ and a holomorphic $H$ on that disc with $\mathrm{Re}H=u_{\phi }$. Then $v_D:=\mathrm{Im}H\circ \phi$ is defined and harmonic on the connected chart disc $D:={\phi }^{-1}(D(\phi (x),r))\subseteq W$, and $u+i{v}_D$ is holomorphic in the chart $\phi$; call such a $D$ a conjugate disc for $u$.

The punctured surface is a legitimate base for covering theory. Let $W:=X\setminus P$ with $P$ finite, and suppose $u$ is harmonic on $W$ with the expansions of part 2. Since $X$ is a connected topological surface, it is locally path-connected and path-connected [F7]. The set $W$ is nonempty (a Riemann surface is nonempty and $P$ is finite, so $X\setminus P\mathrm{\ne }\mathrm{\varnothing }$). $W$ is locally path-connected as an open subset of a locally path-connected space [F7]. $W$ is connected: suppose $W=A\bigsqcup B$ with $A,B$ open in $W$ and nonempty, so $A,B$ are open in $X$ as well; choose for each $p_j$ a chart disc $E_j$ with $E_j\cap P=\{p_j\}$. Each punctured disc $E_j\setminus \{p_j\}$ is connected [F7], and it is contained in $W=A\bigsqcup B$, so it lies entirely in $A$ or entirely in $B$. Put $A^{\ast }:=A\cup \{p_j:E_j\setminus \{p_j\}\subseteq A\},B^{\ast }:=B\cup \{p_j:E_j\setminus \{p_j\}\subseteq B\}\mathrm{.}$ Then $A^{\ast }\cup B^{\ast }=X$, because every point of $W$ lies in $A$ or in $B$ and every $p_j$ lies in $A^{\ast }$ or in $B^{\ast }$, and $A^{\ast }\cap B^{\ast }=\mathrm{\varnothing }$: the two open sets $A,B$ are disjoint, a point $p_j\in A^{\ast }\cap B^{\ast }$ would force the nonempty set $E_j\setminus \{p_j\}$ to lie in both $A$ and $B$, and $A\cap \{p_j:E_j\setminus \{p_j\}\subseteq B\}=\mathrm{\varnothing }$ because that $p_j$ is not in $W$. Each of $A^{\ast },B^{\ast }$ is open in $X$: indeed $A^{\ast }=A\cup \bigcup \{E_j:E_j\setminus \{p_j\}\subseteq A\}$ is a union of open subsets of $X$, since for such $j$ one has $E_j\subseteq A\cup \{p_j\}\subseteq A^{\ast }$, and similarly for $B^{\ast }$. Finally $A\subseteq A^{\ast }$ and $B\subseteq B^{\ast }$ are nonempty. This exhibits $X$ as a disjoint union of two nonempty open sets, contradicting the connectedness of $X$; hence $W$ is connected, and being locally path-connected it is path-connected [F7]. Finally $W$ is semilocally simply connected: every point of $W$ has a chart disc neighbourhood contained in $W$ with simply connected image [F5], which witnesses the condition [F6].

Conjugates on a connected overlap differ by a constant. If $D,D^{\mathrm{\prime }}$ are conjugate discs and $v_D,v_{D^{\mathrm{\prime }}}$ the conjugates from step 1.1, then $v_D-v_{D^{\mathrm{\prime }}}$ is constant on every connected component of $D\cap D^{\mathrm{\prime }}$: on a component $C$, chart expressions of $u+i{v}_D$ and $u+i{v}_{D^{\mathrm{\prime }}}$ are holomorphic, so their difference divided by $i$ is a holomorphic function on an open subset of $\mathbb{C}$ with values in $\mathbb{R}$, hence constant on each chart disc, and these local constants agree on overlaps of chart discs because a locally constant function on the connected set $C$ is constant.

Meridian loops and their normal closure. Fix $x_0\in W$. For each $j$ choose $R_j>0$ with $\overline{D(0,R_j)}\subseteq z_j(U_j)$ and put $V_j:=z_j^{-1}(D(0,R_j))$. These are pairwise disjoint coordinate discs; the prescribed $h_j$ restricts harmonically to $V_j$. Choose $0<r_j<R_j\mathrm{/}2$, put $y_j:=z_j^{-1}(r_j)$, and choose a path ${\lambda }_j$ in $W$ from $x_0$ to $y_j$ [step 1.2]. Let ${\mu }_j(t):=z_j^{-1}(r_j{e}^{2\pi it})$ be the positively oriented meridian based at $y_j$. Set $c_j:=[{\lambda }_j\ast {\mu }_j\ast {\bar{\lambda }}_j]\in {\pi }_1(W,x_0),N:=⟨\text{\hspace{0.17em}⁣}⟨c_1,\dots ,c_n⟩\text{\hspace{0.17em}⁣}{⟩}_{{\pi }_1(W,x_0)}\mathrm{.}$

The period integral of a continuous path. Let $\gamma :[a,b]\to W$ be a continuous path and let $\mathcal{D}$ be the family of all conjugate discs for $u$. The sets ${\gamma }^{-1}(D)$, $D\in \mathcal{D}$, form an open cover of the compact metric space $[a,b]$ [F7], so by the Lebesgue number lemma [F7] there are $a=t_0<t_1<\cdots <t_N=b$ such that each $\gamma ([t_{k-1},t_k])$ lies in some conjugate disc $D_k$ with conjugate $v_k$. Define ${\int }_{\gamma }\beta :={\sum }_{k=1}^N(v_k(\gamma (t_k))-v_k(\gamma (t_{k-1})))\in \mathbb{R}\mathrm{.}$ This is independent of the choice of the $v_k$ for the given partition, because by step 2.1 any two conjugates on $D_k$ and $D_k^{\mathrm{\prime }}$ differ by a constant on the connected component of the overlap containing the connected set $\gamma ([t_{k-1},t_k])$, and that constant cancels in the difference. It is independent of the partition: two admissible data admit a common refinement, the sum over a refinement using the same conjugates as before telescopes to the coarse sum, and two different refinements are compared by step 2.1 on each small interval.

${\pi }_1$ of the punctured chart disc. For each $j$, the composition of the chart $z_j$ with an explicit radial homeomorphism ${\mathbb{C}}^{\times }\to D(0,R_j)\setminus \{0\}$, $w\mapsto R_{j}w\mathrm{/}(1+\mathrm{\mid }w\mathrm{\mid })$, identifies $V_j\setminus \{p_j\}$ homeomorphically with ${\mathbb{C}}^{\times }$, and the circle loop ${\mu }_j$ corresponds to a loop of winding number one; by functoriality of ${\pi }_1$ under homeomorphisms and [F8], ${\pi }_1(V_j\setminus \{p_j\},y_j)$ is infinite cyclic and generated by $[{\mu }_j]$ (regarded there), so its image under the inclusion-induced map is the cyclic subgroup generated by the class of ${\mu }_j$ as a loop in $W$.

The prescribed covering. By step 1.2 the space $W$ satisfies the hypotheses of the subgroup construction [F6], so there is a based connected covering $p:(E,e_0)⟶(W,x_0),p_{\ast }{\pi }_1(E,e_0)=N\mathrm{.}$ The total space $E$ is path-connected: $E$ is connected by construction, it is locally path-connected because $W$ is and [F6], and connected locally path-connected spaces are path-connected [F7].

Elementary properties of the integral. For composable continuous paths $\alpha ,\beta$ one has ${\int }_{\alpha \ast \beta }\beta ={\int }_{\alpha }\beta +{\int }_{\beta }\beta$, for the reversal ${\int }_{\bar{\alpha }}\beta =-{\int }_{\alpha }\beta$, and for a constant path the integral is $0$; these are immediate from the definition of step 3.1 applied to partitions adapted to the concatenation (the value of the constant-path integral is $v(x)-v(x)=0$). Consequently, for a path $\lambda$ from $x$ to $y$ and a loop $\sigma$ at $y$, one has ${\int }_{\lambda \ast \sigma \ast \bar{\lambda }}\beta ={\int }_{\sigma }\beta$.

Homotopy invariance. Let $\gamma ,{\gamma }^{\mathrm{\prime }}:[a,b]\to W$ have the same endpoints and let $H:[a,b]\times [0,1]\to W$ be a homotopy relative to the endpoints joining them. Then ${\int }_{\gamma }\beta ={\int }_{{\gamma }^{\mathrm{\prime }}}\beta$. Indeed, the sets $H^{-1}(D)$ over the conjugate discs cover the compact metric square; by [F7] choose a grid $a=s_0<\cdots <s_M=b$, $0=r_0<\cdots <r_R=1$ so fine that each $H(R_{kl})$ lies in one conjugate disc $D_{kl}$ (use uniform continuity of $H$ and a Lebesgue number). For every grid edge $E$ choose a rectangle $R_{kl}$ containing it and define $\mathrm{inc}(E)$ as the difference of $v_{D_{kl}}$ at the endpoints of $E$, the edge oriented in the increasing first coordinate for horizontal edges and in the increasing second coordinate for vertical edges. Summing over the four boundary edges of a single rectangle $R_{kl}$ telescopes to $0$ because one and the same $v_{D_{kl}}$ is evaluated at the images of the four corners around the closed rectangle. Summing over all rectangles, interior edges occur twice with opposite orientations, and the two contributions agree by step 2.1 applied to the connected image of the edge; hence $0={\sum }_{E\text{ boundary}}\mathrm{inc}(E)$. The two vertical edges contribute $0$, since $H$ fixes the endpoints and the corresponding values are equal, and the bottom and top edges contribute ${\int }_{\gamma }\beta$ and $-{\int }_{{\gamma }^{\mathrm{\prime }}}\beta$. Thus the integrals are equal.

The period homomorphism and its transport. By steps 4.1 and 4.2, for every $x\in W$ the formula ${\phi }_x([\gamma ]):={\int }_{\gamma }\beta$ is a well-defined group homomorphism ${\pi }_1(W,x)\to (\mathbb{R},+)$: it is well defined on classes by step 4.2, additive by step 4.1, and ${\phi }_x([c_x])=0$. For a path $\lambda$ from $x$ to $y$ and a loop $\sigma$ at $y$ one has ${\phi }_x([\lambda \ast \sigma \ast \bar{\lambda }])={\phi }_y([\sigma ])$ by step 4.1.

Simply connected case: all periods vanish. Assume now that $u$ is harmonic on all of $X$ and that $X$ is simply connected. Fix $x_0\in X$. For every loop $\gamma$ at $x_0$ the class $[\gamma ]$ is the identity of ${\pi }_1(X,x_0)$, since that group has exactly one element; hence $\gamma$ is path-homotopic rel endpoints to the constant loop at $x_0$ [F4], so by step 4.2 and step 4.1, ${\phi }_{x_0}([\gamma ])=0$.

A dictionary for the restricted covering. Fix $j$ and let ${\lambda }_j$ be the chosen path from $x_0$ to $y_j$. Let ${\tilde{\lambda }}_j$ be the unique lift of ${\lambda }_j$ starting at $e_0$ and put $e_j:={\tilde{\lambda }}_j(1)\in p^{-1}(y_j)$ [F6]. Let ${\mathrm{\Phi }}_j:{\pi }_1(W,y_j)\to {\pi }_1(W,x_0),{\mathrm{\Phi }}_j([\sigma ]):=[{\lambda }_j\ast \sigma \ast {\bar{\lambda }}_j]$ be transport of classes along ${\lambda }_j$; by [F4] and step 4.1 it is a well-defined group isomorphism, and ${\mathrm{\Phi }}_j([{\mu }_j])=c_j\in N$. I claim $p_{\ast }{\pi }_1(E,e_j)={\mathrm{\Phi }}_j^{-1}(N)\mathrm{.}$ For the inclusion $\subseteq$, let $\beta$ be a loop in $E$ at $e_j$; then ${\tilde{\lambda }}_j\ast \beta \ast {\overline{\stackrel{~}{\lambda }}}_j$ is a loop in $E$ at $e_0$ whose projection is, as a path, the concatenation ${\lambda }_j\ast (p\circ \beta )\ast {\bar{\lambda }}_j$; hence ${\mathrm{\Phi }}_j(p_{\ast }[\beta ])=p_{\ast }[{\tilde{\lambda }}_j\ast \beta \ast {\overline{\stackrel{~}{\lambda }}}_j]\in N$. For the reverse inclusion, let $a\in {\pi }_1(W,y_j)$ with ${\mathrm{\Phi }}_j(a)\in N=p_{\ast }{\pi }_1(E,e_0)$ and choose a loop $\beta$ in $E$ at $e_0$ with $p_{\ast }[\beta ]={\mathrm{\Phi }}_j(a)$; then $\gamma :={\overline{\stackrel{~}{\lambda }}}_j\ast \beta \ast {\tilde{\lambda }}_j$ is a loop in $E$ at $e_j$ whose projection is the loop ${\bar{\lambda }}_j\ast \rho \ast {\lambda }_j$, where $\rho$ is a representative of ${\mathrm{\Phi }}_j(a)=[{\lambda }_j\ast \sigma \ast {\bar{\lambda }}_j]$ for a representative $\sigma$ of $a$. By the cancellation and associativity identities of [F4] the loop ${\bar{\lambda }}_j\ast \rho \ast {\lambda }_j$ represents $[\sigma ]=a$; hence $p_{\ast }[\gamma ]=a$ and $a\in p_{\ast }{\pi }_1(E,e_j)$. Therefore $[{\mu }_j]\in p_{\ast }{\pi }_1(E,e_j)$, because ${\mathrm{\Phi }}_j([{\mu }_j])=c_j\in N$.

Local computation of the periods. Fix $j$ and write $D_j$ for the disc $\mathrm{\mid }{z}_j\mathrm{\mid }<r_j^{\mathrm{\prime }}$ with $r_j^{\mathrm{\prime }}$ chosen so that ${\mu }_j$ runs along $\mathrm{\mid }{z}_j\mathrm{\mid }=r_j<r_j^{\mathrm{\prime }}$ and $D_j\subseteq V_j$. On the disc $D_j$ the function $h_j$ is harmonic, and the field $(-{\mathrm{\partial }}_y{h}_j,{\mathrm{\partial }}_x{h}_j)$ is closed; by [F3] it has a potential $H_j$ on the disc $\mathrm{\mid }{z}_j\mathrm{\mid }<r_j^{\mathrm{\prime }}$ with $\mathrm{\nabla }{H}_j=(-{\mathrm{\partial }}_y{h}_j,{\mathrm{\partial }}_x{h}_j)$ in the chart coordinates, and $h_j+i{H}_j$ is holomorphic there. On the slit disc $S:=D_j\setminus (-\mathrm{\infty },0]$ the principal logarithm of $z_j$ is holomorphic [F9], so $V:=-m_j\arg z_j+H_j$ is a harmonic conjugate of $u$ on $S$: indeed $-m_j\log z_j+h_j+i{H}_j$ is holomorphic on $S$ with real part $-m_j\log \mathrm{\mid }{z}_j\mathrm{\mid }+h_j=u$. Splitting the circle loop ${\mu }_j$ into finitely many arcs of angular width less than $\pi$ and applying the definition of the integral of step 3.1 with rotated logarithm branches from [F9] with continuous argument along each arc, the $H_j$-part telescopes to zero around the closed circle (all increments are those of the single-valued harmonic conjugate $H_j$ along arcs inside $D_j$, so the total change is $0$), while the angular parts add to the total change $2\pi$ of the argument around the positively oriented circle. Hence ${\int }_{{\mu }_j}\beta =-2\pi m_j\mathrm{.}$ By step 4.1 and step 2.2, ${\phi }_{x_0}(c_j)={\int }_{{\mu }_j}\beta =-2\pi m_j$.

A global conjugate by path integration. $X$ is path-connected because it is simply connected [F5]. Define, for $x\in X$, $v(x):={\int }_{\sigma }\beta \text{for any path }\sigma \text{ in }X\text{ from }{x}_0\text{ to }x\mathrm{.}$ This is well defined: two paths $\sigma ,{\sigma }^{\mathrm{\prime }}$ from $x_0$ to $x$ give ${\int }_{\sigma }\beta -{\int }_{{\sigma }^{\mathrm{\prime }}}\beta ={\int }_{\sigma \ast {\bar{\sigma }}^{\mathrm{\prime }}}\beta =0$ by steps 4.1 and 5.2, since $\sigma \ast {\bar{\sigma }}^{\mathrm{\prime }}$ is a loop at $x_0$.

Each component over a punctured disc is one-sheeted. Let $D_j^{\times }:=V_j\setminus \{p_j\}$ and restrict $p$ over this open set. By step 3.2 its fundamental group is generated by the meridian $[{\mu }_j]$, whose image in ${\pi }_1(W,y_j)$ lies in $p_{\ast }{\pi }_1(E,e_j)$ by step 5.3. The lifting criterion [F6] gives a section of the restricted covering through $e_j$; the component $Z$ containing $e_j$ therefore has a surjective map on fundamental groups and is one-sheeted by the sheet-index formula [F6]. Now let $e$ be any other point above $y_j$. Choose a path in the connected space $E$ from $e_j$ to $e$, and write $g$ for the class of its projected loop at $y_j$. Path conjugation identifies $p_{\ast }{\pi }_1(E,e)$ with $g^{-1}{p}_{\ast }{\pi }_1(E,e_j)g$. By step 5.3 the latter subgroup is ${\mathrm{\Phi }}_j^{-1}(N)$, which is normal because $N$ is a normal closure; hence $[{\mu }_j]$ also lies in $p_{\ast }{\pi }_1(E,e)$. The same section and sheet-index argument makes the component through $e$ one-sheeted. Every component of a covering over the connected, locally path-connected disc $D_j^{\times }$ contains a point above $y_j$, so every component maps homeomorphically onto $D_j^{\times }$. The total preimage may have several components.

$v$ is a global harmonic conjugate. Let $x\in X$, let $D\ni x$ be a conjugate disc with conjugate $v_D$, and let $y\in D$. Choose a path $\sigma$ from $x_0$ to $x$ and a path $\tau$ inside $D$ from $x$ to $y$; then, using step 3.1 with the single conjugate $v_D$ on the second piece, $v(y)={\int }_{\sigma \ast \tau }\beta =v(x)+(v_D(y)-v_D(x))$. Hence $v=v_D+\text{constant}$ on $D$; so $v$ is harmonic on $D$ and $u+iv$ is holomorphic in the coordinate of $D$, up to the additive imaginary constant. As the conjugate discs cover $X$, $v$ is a harmonic conjugate of $u$ on $X$, and part 1 of the statement follows.

Extending the covering across the punctures. For each $j$ and each component $C$ of $p^{-1}(D_j^{\times })$, the homeomorphism $p{\mathrm{\mid }}_C:C\to D_j^{\times }$ extends to a homeomorphism $C\cup \{c_C\}\to V_j$ of the one-point extensions, sending the new point $c_C$ to $p_j$. Form $E^{\mathrm{\prime }}:=E\cup \{c_C\}$ over all $j$ and components $C$, with the topology generated by open sets of $E$ and sets $\{c_C\}\cup (p{\mathrm{\mid }}_C{)}^{-1}(z_j^{-1}(D(0,t))\setminus \{p_j\})$, for $0<t<R_j$, and extend $p$ to $p^{\mathrm{\prime }}:E^{\mathrm{\prime }}\to X$ by $p^{\mathrm{\prime }}(c_C):=p_j$. Then $p^{\mathrm{\prime }}$ is a covering map: over $W$ it restricts to the covering $p$, and over each $V_j$ the preimage is the disjoint union of the spaces $C\cup \{c_C\}\cong V_j$, each mapping homeomorphically onto $V_j$, so the local triviality conditions hold at interior points and at the added points. Moreover $E^{\mathrm{\prime }}$ is connected: $E$ is connected by step 3.3, and each added point $c_C$ lies in the closure of $C\setminus \{c_C\}\subseteq E$, so $E^{\mathrm{\prime }}$ is the union of $E$ with points in its closure.

Conclusion: the meridians generate. The map $p^{\mathrm{\prime }}:E^{\mathrm{\prime }}\to X$ is a connected covering of the simply connected locally path-connected space $X$, hence is one-sheeted and an isomorphism [F6]. Therefore $p:E\to W$ is one-sheeted, so $p_{\ast }:{\pi }_1(E,e_0)\to {\pi }_1(W,x_0)$ is an isomorphism, and its image is $N$; hence ${\pi }_1(W,x_0)=N=⟨\text{\hspace{0.17em}⁣}⟨c_1,\dots ,c_n⟩\text{\hspace{0.17em}⁣}⟩\mathrm{.}$ In particular every element of ${\pi }_1(W,x_0)$ is a finite product of conjugates of the classes $c_j^{\pm 1}$ [F10].

All periods are integral multiples of $2\pi$. By step 8.1 every class in ${\pi }_1(W,x_0)$ is a finite product of conjugates of the $c_j^{\pm 1}$; since ${\phi }_{x_0}$ is a homomorphism [step 5.1] and ${\phi }_{x_0}(g{c}_j{g}^{-1})={\phi }_{x_0}(c_j)=-2\pi m_j$ for every $g$, while ${\phi }_{x_0}(c_j^{-1})=2\pi m_j$, every period ${\phi }_{x_0}([\gamma ])$ lies in $2\pi \mathbb{Z}$.

Construction of $F$. Define, for $x\in W$, $F(x):=e^{-u(x)}\exp (-i{\int }_{\sigma }\beta )\text{for any path }\sigma \text{ in }W\text{ from }{x}_0\text{ to }x\mathrm{.}$ This is well defined: two paths $\sigma ,{\sigma }^{\mathrm{\prime }}$ differ by a loop at $x_0$ and the two exponents differ by an element of $2\pi \mathbb{Z}$ by step 9.1, so the exponentials agree.

$F$ is holomorphic and nonvanishing with $\mathrm{\mid }F\mathrm{\mid }=e^{-u}$. On a conjugate disc $D$ containing a point $x$, and for $y\in D$, the path $\sigma$ from $x_0$ to $x$ followed by a path in $D$ from $x$ to $y$ computes $\int \beta ={\int }_{\sigma }\beta +(v_D(y)-v_D(x))$, so $F(y)=F(x)e^{-(u(y)+i{v}_D(y))}{e}^{u(x)+i{v}_D(x)}$ on $D$. Since $u+i{v}_D$ is holomorphic in the chart of $D$, the right-hand side exhibits $F$ as a holomorphic function on $D$ with $F(y)\mathrm{\ne }0$; as the conjugate discs cover $W$, $F$ is holomorphic on $W$ and vanishes nowhere. Taking moduli gives $\mathrm{\mid }F(y)\mathrm{\mid }=e^{-u(y)}>0$ everywhere on $W$.

Local form at each puncture. Fix $j$. On $W$ define $G:=z_j^{-m_j}F$ near $p_j$, where $z_j^{-m_j}$ is the holomorphic nonvanishing power function of [F9] on the punctured chart; then $G$ is holomorphic on $D_j\setminus \{p_j\}$. To compute its size, note that $\mathrm{\mid }F\mathrm{\mid }=e^{-u}=e^{m_j\log \mathrm{\mid }{z}_j\mathrm{\mid }-h_j}=\mathrm{\mid }{z}_j{\mathrm{\mid }}^{m_j}{e}^{-h_j}$ by step 11.1 and the expansion of $u$; hence $\mathrm{\mid }G\mathrm{\mid }=e^{-h_j}$ on $D_j\setminus \{p_j\}$, and $h_j$ is continuous on the disc $D_j$ with $p_j$ in its interior, so $G$ is bounded near $p_j$. By the removable-singularity characterizations [F9], $G$ extends holomorphically over $p_j$, with $\mathrm{\mid }G(p_j)\mathrm{\mid }={\lim }_{\mathrm{\mid }{z}_j\mathrm{\mid }\to 0}{e}^{-h_j}=e^{-h_j(p_j)}>0$. Writing $G_j$ for the extension gives $F=z_j^{m_j}{G}_j\text{near }{p}_j,G_j\text{ holomorphic at }0,G_j(0)\mathrm{\ne }0.$

The extension is meromorphic with the asserted divisor. By step 12.1, at each puncture the chart expression of $F$ is either holomorphic at the centre (if $m_j\ge 0$, with a zero of order exactly $m_j$ when $m_j>0$ and a nonzero value when $m_j=0$) or has a pole of order $-m_j$ there when $m_j<0$; in the latter case the reciprocal chart expression is holomorphic at the centre with value $0$, which is exactly the chart condition for holomorphy of a map $X\to \widehat{\mathbb{C}}$ in the chart ${\varphi }_{\mathrm{\infty }}$ [F9]. Hence $F$ extends to a holomorphic map $F:X\to \widehat{\mathbb{C}}$, and it is not the constant map $\mathrm{\infty }$ because $F(W)\subseteq {\mathbb{C}}^{\times }$ by step 11.1; so $F$ is meromorphic [F9]. Orders are those of The order of a zero of a holomorphic function, and the pole order is read off from the reciprocal as in Isolated singularities: removable, poles, and essential singularities.

No other zeros or poles. On $W=X\setminus P$ the modulus $\mathrm{\mid }F\mathrm{\mid }=e^{-u}$ is finite and strictly positive by step 11.1, so $F$ has neither zeros nor poles in $W$; together with step 13.1 the zeros and poles of $F$ are exactly the points $p_j$ with $m_j\mathrm{\ne }0$, with the orders $\mathrm{\mid }{m}_j\mathrm{\mid }$ and the signs described. This completes the proof of part 2, and part 1 was proved in step 7.1.