Weak solutions of a degree-zero divisor and the logarithmic-derivative identity

For a path contained in a coordinate disk $V$ identified with the unit disk, write its endpoints as $a,b$ and choose $0<r<r^{\mathrm{\prime }}<1$ with the entire compact path image in $\{\mathrm{\mid }z\mathrm{\mid }<r\}$. Choose $\psi$ equal to $1$ on $\{\mathrm{\mid }z\mathrm{\mid }\le r\}$ and supported in $\{\mathrm{\mid }z\mathrm{\mid }<r^{\mathrm{\prime }}\}$ by [F3]. If $a=b$, set $f_0=1$; the integral of a closed form on this path is zero by [F1]. If $a\mathrm{\ne }b$, put $R(z)=(z-b)\mathrm{/}(z-a)$ and use $L=\mathrm{Log}R$ outside the segment $[a,b]$. Indeed $R(z)$ belongs to the nonpositive real ray only on $[a,b]$, which lies inside $\{\mathrm{\mid }z\mathrm{\mid }<r\}$, so [F2] defines $L$ throughout the annulus $r<\mathrm{\mid }z\mathrm{\mid }<1$. Set $f_0=R$ for $\mathrm{\mid }z\mathrm{\mid }\le r$ and $f_0=\exp (\psi L)$ for $r<\mathrm{\mid }z\mathrm{\mid }<1$. These formulas agree smoothly across $\mathrm{\mid }z\mathrm{\mid }=r$: where $L$ is defined, $f_0=R\exp ((\psi -1)L)$, and $\psi -1$ vanishes smoothly on the inner disk. Since $f_0=1$ near $\mathrm{\partial }V$, extend it by $1$ off $V$. This gives a weak solution of $Q-P$ with a simple zero at $b$, a simple pole at $a$, and no other zeros or poles.

For claim 3, let $f$ and $g$ be weak solutions of $D$. Near a support point both have the local form $z^{n_p}{h}_f$ and $z^{n_p}{h}_g$ with the same integer $n_p=D(p)$ and smooth nowhere-vanishing factors, so $g\mathrm{/}f=h_g\mathrm{/}{h}_f$ is smooth and nowhere vanishing near $p$; away from $\mathrm{\mid }D\mathrm{\mid }$ both functions are smooth and nowhere vanishing. Hence $g\mathrm{/}f$ is smooth and nowhere vanishing on all of $X$.

The endpoint integral of [F1] is invariant under fixed-endpoint homotopy: subdivide the compact homotopy square into sufficiently small triangles in primitive neighborhoods, where every boundary sum telescopes, and cancel the interior edges. A coordinate disk contracts to a point, so this makes integration from a fixed point path independent on the disk. The resulting function is a local primitive plus a constant near each point, hence a smooth global primitive of any closed form there. Let $\omega =dg$ near the closed disk $\mathrm{\mid }z\mathrm{\mid }\le r^{\mathrm{\prime }}$, and multiply the local primitive by a cutoff supported in $V$ and equal to $1$ near that disk, obtaining a global smooth $g$ without changing $dg$ where $d{f}_0\mathrm{/}{f}_0$ is supported. Put $\alpha =d{f}_0\mathrm{/}{f}_0$ away from $a,b$. It is closed there, since $d(f_0^{-1}d{f}_0)=-f_0^{-2}d{f}_0\wedge d{f}_0=0$. The local form in [F4] shows that $\alpha \wedge dg$ is absolutely integrable: its coefficient is $O(1\mathrm{/}\mathrm{\mid }z-a\mathrm{\mid })+O(1\mathrm{/}\mathrm{\mid }z-b\mathrm{\mid })$. On a compact coordinate disk containing its support with small disks about $a,b$ deleted, $d(g\alpha )=dg\wedge \alpha =-\alpha \wedge dg$. The outer boundary contributes zero since $\alpha =0$ there. On the two inner circles the boundary orientation is clockwise; [F4] therefore gives $\int \alpha \wedge dg=2\pi i(g(b)-g(a))$ as their radii tend to zero. Thus $\frac{1}{2\pi i}{\int }_X(d{f}_0\mathrm{/}{f}_0)\wedge \omega =g(b)-g(a)={\int }_{\gamma }\omega$ by [F1]. For $a=b$ both sides vanish.

Subdivide each curve into finitely many subpaths contained in coordinate disks, using compactness and the Lebesgue-number argument of [F1], and construct the factor $f_{j\mathrm{\ell }}$ for each subpath by step 1.1. Each construction depends only on its subpath and disk, not on $\omega$; every closed form has a primitive on such a disk, since local primitive endpoint integrals are invariant under fixed-endpoint homotopy by subdividing a compact homotopy square into primitive neighborhoods, whose boundary increments telescope. Form the finite product $f={\prod }_{j,\mathrm{\ell }}{f}_{j\mathrm{\ell }}$. At any endpoint of the subdivided curves, each factor has an integer coordinate power times a smooth unit; changing a centered holomorphic coordinate multiplies that power by a holomorphic unit. Summing the endpoint exponents gives exactly the boundary of the original chain: intermediate endpoint contributions cancel, even when endpoints repeat or a subpath is closed. Therefore $f=z^{D(p)}h$ with $h$ smooth and nowhere zero at support points, and $f$ extends smoothly and nonvanishingly at all canceled intermediate endpoints. Away from the endpoints it is smooth and nonzero. On their complement, the finite product rule gives $df\mathrm{/}f={\sum }_{j,\mathrm{\ell }}d{f}_{j\mathrm{\ell }}\mathrm{/}{f}_{j\mathrm{\ell }}$.

Sum the absolutely convergent model identities from step 2.1 and use the product rule of step 2.2 and finite additivity of the path integral in [F1]. This gives $\frac{1}{2\pi i}{\int }_X(df\mathrm{/}f)\wedge \omega ={\sum }_{j,\mathrm{\ell }}{\int }_{{\gamma }_{j\mathrm{\ell }}}\omega ={\int }_c\omega$ for every closed smooth complex $1$-form. For holomorphic $\omega$, the $(1,0)$ term wedges to zero by [F7], while the local form $f=z^{D(p)}h$ gives $\bar{\mathrm{\partial }}f\mathrm{/}f=\bar{\mathrm{\partial }}h\mathrm{/}h$, a smooth form across every support point. Hence the displayed holomorphic identity has an ordinary smooth top-form integral.

The construction proves the weak-solution and integral claims in steps 2.2 and 3.1, and step 1.2 proves uniqueness up to a smooth nowhere-vanishing factor. The zero chain gives the empty product $f=1$; a nonzero closed chain can instead give a nonconstant nowhere-zero $f$, as its period identity requires. All selections in the construction are finite, and full AC covers the inherited Stokes hypothesis in [F9].