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Let $\wp$ be the Weierstrass elliptic function. Is there a meromorphic function $f$ from $\mathbb{C}/L$ such that $f'(z) = \wp(z)$? Here $L$ is the usual lattice $(m\omega_1 + n\omega_2 | m,n \in \mathbb{Z})$.

As a hint I am given "What would the poles of $f$ look like?" but I have no idea how to think on this.

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    $\begingroup$ The poles of $\wp$ look like $1/z^2$, so the poles of $f$ would look like $1/z$. $\endgroup$
    – mr_e_man
    Commented May 7, 2019 at 20:37
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    $\begingroup$ Could you just integrate the series definition termwise? $$\wp(z)=\sum_{\omega\in L}\bigg(\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\bigg)$$ $$f(z)=\sum_{\omega\in L}\bigg(\frac{-1}{z-\omega}-\frac{z}{\omega^2}\bigg)$$ $\endgroup$
    – mr_e_man
    Commented May 7, 2019 at 20:45

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This is the negative of the Weierstrass zeta function. It has simple poles with residue 1 at each point of the lattice. It is not an elliptic function.

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