Franklin's Notes

Representation of meromorphic periodic functions

Suppose that $f$ is a meromorphic function that is also periodic, say, with period $1$, so that $f(z+1)=f(z)$ for all $z\in\mathbb C$ for which it is defined. Then we may define a single-valued meromorphic function $g$ on $\mathbb C$ as follows: Note that the integral involved in this expression may take on several different values depending on the path of integration from $z_0$ to $z$. However, as proven here , these values all differ by integer multiples of $2\pi i$, meaning that the above actually defines $g$ unambiguously as a function of $z$. Let us set $z_0=1$, and write $z=e^{2\pi i w}$ for $z\ne 0$. Then we have that Depending on the path of integration, this integral can take on values of the form $2\pi i w + 2\pi i n$ for $n\in\mathbb Z$, so that we have Hence, we may express the function $f$ in the following form: where $g$ is also meromorphic.

More generally, if $f$ is a meromorphic function with period $\omega$, we have that $f(z\omega)$ has period $1$, so that $f(z\omega)=g(e^{2\pi i z})$ for some meromorphic function $g$, and therefore giving us an analogous way of representing periodic functions with an arbitrary period.


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