Cauchy's integral formula for a null-homologous cycle

Given: An open $\mathrm{\Omega }$, a holomorphic $f:\mathrm{\Omega }\to \mathbb{C}$, and a cycle $\mathrm{\Gamma }$ with ${\mathrm{\Gamma }}^{\ast }\subseteq \mathrm{\Omega }$ which is null-homologous in $\mathrm{\Omega }$.

With $g$ the filled difference quotient of $f$, the function $h$ 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 ${\mathrm{\Omega }}_0=\{z\mathrm{\notin }{\mathrm{\Gamma }}^{\ast }:n(\mathrm{\Gamma },z)=0\}$ is a well-defined entire function; it is bounded, and for every $\epsilon >0$ there is $R>0$ with $\mathrm{\mid }h(z)\mathrm{\mid }<\epsilon$ whenever $\mathrm{\mid }z\mathrm{\mid }>R$ (Dixon's glued function is entire and vanishes at infinity).

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

If $f$ is continuous on the trace of a complex chain, then ${\int }_{\mathrm{\Gamma }}fdz={\sum }_{k<r,m_k\mathrm{\ne }0}{m}_k{\int }_{{\gamma }_k}fdz$; 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 with trace the union of the ${\gamma }_k^{\ast }$ having $m_k\mathrm{\ne }0$ (Complex chains, their traces, and cycles).

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

Complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand); finite sums in the additive commutative monoid of $\mathbb{C}$ are additive, and complex-field distributivity permits scaling term by term (A finite sum in a commutative monoid indexed by an arbitrary finite set, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$).

The filled difference quotient $g$ of a holomorphic $f$ on $\mathrm{\Omega }$ equals $(f(\zeta )-f(z))\mathrm{/}(\zeta -z)$ off the diagonal and $f^{\mathrm{\prime }}(z)$ on it (The filled difference quotient of a holomorphic function is jointly continuous).

A function holomorphic on all of $\mathbb{C}$ is entire (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

A holomorphic function is continuous (Complex differentiability at a point implies continuity there).