19. Application 2.13.1 - zeta function.

Let $f$ be an elliptic function, whose lattice is given by $L = ω_{1} Z + ω_{2} Z$ . Let $a_{1}, \dots, a_{n}$ be it's poles in a cell $P$ , whose principal parts are given by:

f (z) = \sum_{m = 1}^{r_{k}} \frac{c_{m}^{(k)}}{(z - a_{k})^{m}} + O (1)

Then we can write $f$ in the following way in terms of 1;;the $ζ$ -function

f (z) = c o n s t + \sum_{k = 1}^{n} \sum_{m = 1}^{r_{k}} c_{m}^{(k)} \frac{(- 1)^{m - 1}}{(m - 1)!} ζ^{(m - 1)} (z - a_{k})

where $ζ^{(m - 1)}$ is the $m - 1$ th derivative.

PROOF:

Look at the Laurant series around each $a_{k}$ and show they have the same principal part. Then by Louiville we are done.
We do need to show that the RHS is elliptic. To do this, we use that sum of all residues of an elliptic function is zero.