Dixon's gluing traced on the boundary cycle of an annulus

Given: The sets and contours above, with $f(z)=1\mathrm{/}z$.

If $\mathrm{\Omega }$ is open, $f$ is holomorphic on $\mathrm{\Omega }$, and $\mathrm{\Gamma }$ is a null-homologous cycle with trace in $\mathrm{\Omega }$, then, with $g$ the filled difference quotient of $f$, the function equal to $(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}g(\zeta ,z)d\zeta$ on $\mathrm{\Omega }$ and to $(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}f(\zeta )(\zeta -z{)}^{-1}d\zeta$ on ${\mathrm{\Omega }}_0$ is a well-defined entire function, bounded and tending to $0$ at infinity (Dixon's glued function is entire and vanishes at infinity). The filled difference quotient is $(f(\zeta )-f(z))\mathrm{/}(\zeta -z)$ off the diagonal and $f^{\mathrm{\prime }}(z)$ on it (The filled difference quotient of a holomorphic function is jointly continuous).

For a cycle null-homologous in an open $\mathrm{\Omega }$ and $f$ holomorphic there, $n(\mathrm{\Gamma },z)f(z)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}f(\zeta )(\zeta -z{)}^{-1}d\zeta$ for every $z\in \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\ast }$ (Cauchy's integral formula for a null-homologous cycle).

For $p\in \mathbb{C}$ and $0<r_1<r_2$ the chain built from the positively oriented circles of radii $r_2$ and $r_1$ about $p$ with coefficients $+1$ and $-1$ is a cycle with trace the two circles, index $0$ for $\mathrm{\mid }z-p\mathrm{\mid }<r_1$, $1$ for $r_1<\mathrm{\mid }z-p\mathrm{\mid }<r_2$ and $0$ for $\mathrm{\mid }z-p\mathrm{\mid }>r_2$; it is null-homologous in $\{s_1<\mathrm{\mid }z-p\mathrm{\mid }<s_2\}$ whenever $0<s_1<r_1$ and $r_2<s_2$ (The boundary cycle of a round annulus has index $1$ inside the annulus and $0$ on either side).

A cycle with trace in an open $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when its index vanishes at every point outside $\mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

${\int }_{\mathrm{\Gamma }}fdz={\sum }_{k<r,m_k\mathrm{\ne }0}{m}_k{\int }_{{\gamma }_k}fdz$, and for $z\mathrm{\notin }{\mathrm{\Gamma }}^{\ast }$ one has $n(\mathrm{\Gamma },z)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}d\zeta \mathrm{/}(\zeta -z)$ (Integration over a complex chain and the index of a chain); a chain is a finite list of integer-weighted contours (Complex chains, their traces, and cycles).

For $a\in \mathbb{C}$, $r>0$ and every integer $m$, the positively oriented circle $\gamma (t)=a+r\exp (it)$ on $[0,2\pi ]$ satisfies ${\int }_{\gamma }(z-a{)}^{m}dz=2\pi i$ when $m=-1$ and $0$ otherwise (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

Chain integration and the index are additive in the chain and reverse with it (Chain integration and the index are additive in the chain, and reverse with it); complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand).

For $a\in \mathbb{C}$, $r>0$ and $k\in \mathbb{Z}$, the contour $a+r\exp (ikt)$ on $[0,2\pi ]$ has index $k$ for $\mathrm{\mid }z-a\mathrm{\mid }<r$ and $0$ for $\mathrm{\mid }z-a\mathrm{\mid }>r$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

Every bounded entire function is constant (Liouville's theorem: every bounded entire function is constant).

Nonvanishing quotients of functions complex differentiable at a point are complex differentiable there (Linearity, product, reciprocal, and quotient rules for complex derivatives).