Every cycle in a connected plane domain is null-homologous in that domain

Given: The annulus $A=\{z:\frac{1}{2}<\mathrm{\mid }z\mathrm{\mid }<2\}$ and the contour $C(t)=\exp (it)$ on $[0,2\pi ]$, taken as the chain with the single term $(1,C)$.

A cycle $\mathrm{\Gamma }$ with trace in an open $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when $n(\mathrm{\Gamma },q)=0$ for every $q\in \mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

A complex domain is homologically simply connected when every cycle with trace in it is null-homologous in it (Homologically simply connected complex domains).

The annulus $\{\frac{1}{2}<\mathrm{\mid }z\mathrm{\mid }<2\}$ is a complex domain, and the unit circle $C$ in it is a cycle with $n(C,0)=1$, so $C$ is not null-homologous in that annulus (A connected plane domain that is not homologically simply connected).

For $a\in \mathbb{C}$, $r>0$ and $k\in \mathbb{Z}$, the contour $a+r\exp (ikt)$ on $[0,2\pi ]$ is a closed complex contour with 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).

If $\mathrm{\Gamma }$ is a cycle with trace in an open $\mathrm{\Omega }$, null-homologous in $\mathrm{\Omega }$, and $f$ is holomorphic on $\mathrm{\Omega }$, then ${\int }_{\mathrm{\Gamma }}fdz=0$ (Cauchy's theorem for a null-homologous cycle).

For a positively oriented circle $a+r\exp (it)$ with $r>0$, $(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-a)=1$ (The normalized integral around a positively oriented circle centred at a is 1).

A single closed contour with coefficient $1$ is a cycle whose trace is that contour's trace (Complex chains, their traces, and cycles) and whose index is that contour's winding number (Integration over a complex chain and the index of a chain).

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