---- > [!definition] Definition. ([[function on the (unit) circle]]) > We say that a $2\pi$-[[periodic]] function on $\rr$ (e.g., $f(\theta)=e ^{in \theta}$) is a function **on the (unit) circle**. > [!justification] > The connection arises as follows. A point on the [[unit circle]] takes the form $e ^{i \theta }$, where $\theta \in \rr$ is unique up to integer multiples of $2\pi$. If $F$ is a function on the [[unit circle]] ($\dom F=$ $\{z \in \cc : | z |=1\}$), we can define a function $f$ on $\rr$ by $f(\theta):=F(e ^{i \theta}), \ \theta \in \rr$ and observe that $f$ must be $2\pi$-[[periodic]] on $\rr$. \ Since $f$ has [[periodic|period]] $2\pi$, we may restrict to to any [[interval]] of length $2\pi$, say $[0,2\pi]$, and still capture the initial function $F$ on the circle. Of course, $f$ must take on the same values at its start and end points since these corresponding to the same point on the circle. Conversely, any function on $[0,2\pi]$ for which $f(0)=f(2\pi)$ can be extended to a $2\pi$-[[periodic]] function on $\rr$, which can then be identified with a function on the circle. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```