Every cycle in a round annulus has one period, that of the central circle

Given: Radii $0<r_1<\rho <r_2$, the annulus $A$, the circle $C$, and a cycle $\mathrm{\Gamma }$ with trace in $A$.

If $f$ is holomorphic on an open $\mathrm{\Omega }$ and two cycles with traces in $\mathrm{\Omega }$ are homologous in $\mathrm{\Omega }$, their integrals of $f$ agree (Holomorphic integrals agree on homologous cycles).

Two cycles with traces in $\mathrm{\Omega }$ are homologous in $\mathrm{\Omega }$ exactly when their indices agree at every point of $\mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

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

For a cycle $\mathrm{\Gamma }$ the trace is compact, the index is constant on every connected component of $\mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$, each such component is open, and there is $R>0$ with $n(\mathrm{\Gamma },p)=0$ whenever $\mathrm{\mid }p\mathrm{\mid }>R$ (The index of a cycle is locally constant off its trace and vanishes far from it).

For $a\in \mathbb{C}$, $r>0$ and every integer $m$, the positively oriented circle $a+r\exp (it)$ 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).

${\int }_{\mathrm{\Gamma }}fdz={\sum }_{k^{\mathrm{\prime }}<r,m_{k^{\mathrm{\prime }}}\mathrm{\ne }0}{m}_{k^{\mathrm{\prime }}}{\int }_{{\gamma }_{k^{\mathrm{\prime }}}}fdz$ and $n(\mathrm{\Gamma },p)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}dz\mathrm{/}(z-p)$ (Integration over a complex chain and the index of a chain), a chain being a finite list of integer-weighted contours, and a one-term chain carried by a closed contour being a cycle (Complex chains, their traces, and cycles).

For $c\in \mathbb{C}$ and every real $R^{\mathrm{\prime }}>0$, the set $\{z:\mathrm{\mid }z-c\mathrm{\mid }\ge R^{\mathrm{\prime }}\}$ is path-connected and connected (The exterior of a closed disc in the plane is path-connected).

The connected component of a point is the union of all connected subsets containing it (Connected components, quasicomponents, and totally disconnected spaces) and contains every connected subset containing that point (The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

A set is convex when it contains the segment between any two of its points (A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points); a subset joined by paths inside it is path-connected (Paths, path-connected spaces and path components) and hence connected (Every path-connected space is connected, and every path component lies inside a component).

$\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$ 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); a subset is bounded when it is empty or lies inside some ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

The index of a cycle about a point off its trace is an integer (The index of a cycle about a point off its trace is an integer).

Constants and the identity are holomorphic, and nonvanishing quotients of holomorphic functions are holomorphic (Linearity, product, reciprocal, and quotient rules for complex derivatives).