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Suppose we have a lattice $\Lambda.$ We have the Laurent expansion of $p$ is: $$p(z) = \frac{1}{z^2} + \sum_{n = 2, n \text{ even}} (n + 1)G_{n + 2}(\Lambda)z^n $$ where $G_{n + 2}(\Lambda) = \sum_{w \in \Lambda \setminus 0} \frac{1}{w^{n + 2}}$ and for all $z$ such that $0 < |z| < \inf\{|w|:w \in \Lambda \setminus 0\}.$

The book I am reading states that $(p'(z))^2 = 4p(z)^3 - g_2(\Lambda)p(z) - g_3(\Lambda)$ where $g_2(\Lambda) = 60G_4(\Lambda), g_3(\Lambda) = 140G_4(\Lambda).$ To prove this, they use that $p(z) = \frac{1}{z^2} + 3G_4(\Lambda)z^2 + 5G_6(\Lambda)z^4 + O(z^6)$ and $p'(z) = -\frac{2}{z^3} + 6G_4(\Lambda)z + 20G_6(\Lambda)z^3 + O(z^5).$ However, isn't this using the Laurent series expansion which is only valid in a small annulus centered at the origin? I do know that $p(z)$ is $\Lambda$-periodic, but how can we justify that this expansion is valid in just the fundamental parallelogram of our lattice $\Lambda.$ This seems unlikely. If our lattice is say spanned by a small $w_1$ and a huge $\omega_2.$ Any guidance is appreciated.

green frog
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It doesn't matter. By uniqueness of meromorphic continuation, if you can prove a relationship between two meromorphic functions in any open set (such as a neighbourhood or deleted neighbourhood of a point), no matter how small, it will hold throughout the entire region where the functions are meromorphic. Since you presumably already know that $\wp$ (and therefore also $\wp'$) is meromorphic on $\mathbb{C}$, the result holds on the whole plane. Whether you include the points that are poles or not is a different in different treatments of the subject, but the poles form a countable discrete set and so are reasonably unimportant here, in that they cannot form a genuine obstruction.

(By the way, the Laurent expansions are valid on a punctured disk surrounding $0$, rather than a nondegenerate annulus, since the principal part of the Laurent expansion contains only finitely many terms.)

Chappers
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