Dixon's glued function is entire and vanishes at infinity

Given: An open $\mathrm{\Omega }$, a holomorphic $f:\mathrm{\Omega }\to \mathbb{C}$, and a cycle $\mathrm{\Gamma }={\sum }_{k<r}{m}_k{\gamma }_k$ with ${\mathrm{\Gamma }}^{\ast }\subseteq \mathrm{\Omega }$ which is null-homologous in $\mathrm{\Omega }$; the plane carries the Euclidean metric of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves.

If $\gamma$ is a rectifiable contour, $W$ is open and $\phi$ is continuous on ${\gamma }^{\ast }\times W$ with $\phi (w,\cdot )$ holomorphic on $W$ for each $w\in {\gamma }^{\ast }$, then $z\mapsto {\int }_{\gamma }\phi (\zeta ,z)d\zeta$ is holomorphic on $W$ (A contour integral of a jointly continuous, parameter-holomorphic integrand is holomorphic).

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, and is continuous on $\mathrm{\Omega }\times \mathrm{\Omega }$ (The filled difference quotient of a holomorphic function is jointly continuous).

For each fixed $z\in \mathrm{\Omega }$ the map $\zeta \mapsto g(\zeta ,z)$ is holomorphic on $\mathrm{\Omega }$, and for each fixed $\zeta \in \mathrm{\Omega }$ the map $z\mapsto g(\zeta ,z)$ is holomorphic on $\mathrm{\Omega }$ (The filled difference quotient is holomorphic in each variable separately).

For a complex chain $\mathrm{\Gamma }$ and $\phi$ continuous on ${\mathrm{\Gamma }}^{\ast }$, the function $z\mapsto (2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}\phi (\zeta )(\zeta -z{)}^{-1}d\zeta$ is holomorphic on $\mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$ (The Cauchy transform of a cycle is holomorphic off its trace, with the expected derivatives).

For a cycle $\mathrm{\Gamma }$ the trace ${\mathrm{\Gamma }}^{\ast }$ is compact, the index is locally constant on $\mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$, the set ${\mathrm{\Omega }}_0$ of points off the trace where the index vanishes is open, and there is $R_1>0$ with $n(\mathrm{\Gamma },p)=0$ whenever $\mathrm{\mid }p\mathrm{\mid }>R_1$ (The index of a cycle is locally constant off its trace and vanishes far from it).

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).

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

If $\mathrm{\mid }f(z)\mathrm{\mid }\le M$ on the trace of a rectifiable contour $\gamma$, with $M\ge 0$, then $\mathrm{\mid }{\int }_{\gamma }fdz\mathrm{\mid }\le ML(\gamma )$ (ML estimate: a contour integral is bounded by a supremum bound times path length); complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand).

A compact subset is closed and bounded (A compact subset of a metric space is closed and bounded); a continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a finite union of compact subsets is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact); a closed bounded interval and a closed disc are compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

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

A set is closed exactly when its complement is open, and a set is open exactly when each of its points admits a ball inside it (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space); a set is bounded when it is empty or lies inside a ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

$\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$ and $\mathrm{\mid }z+w\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

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 $(a-bi)\mathrm{/}(a^2+b^2)$).

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

Finite linear combinations of holomorphic functions are holomorphic (Linearity, product, reciprocal, and quotient rules for complex derivatives).