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Consider the elliptic function, $\rho(z) = \frac{1}{z^{2}} + \sum(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}) $ here $m,n \neq 0,0 $. The task is to prove this function elliptic, so doubly periodic with periods 1 and w and also meropmorphic with a double pole at m + nw (m * 1 + n * w) for all complex m,n. Anyone has any ideas on how should I proceed? Thanks.

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The simpler function f(z) = $\sum_{w\,in,\Lambda}\frac{1}{(z-w)^3}$ is $\Lambda$-elliptic for all lattice $\Lambda$, odd and of order 3. However the theory states that all elliptic function has an order at least 2 and the desired function satisfying this minimal property has the more complicated Weirstrass form $\wp(z) = \frac{1}{z^{2}} + \sum(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}})$ in wich the$\Sigma$ refers actually to a double summation and the second term, without z, has no other purpose than to ensure the necessary convergence (the first term diverges). Note that this pole of order two can be given as a single pole of order two or two simple poles. In the first case you have the Weirstrass elliptic functions $\wp(z)$ and in the second the Jacobi elliptic functions. The great importance of the functions $\wp(z)$ is that, being both transcendent, there is a relationship of algebraic dependency between $\wp(z)$ and its derivative which has the form of a cubic equation giving precisely the Weirstrass form of an elliptic curve.

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