I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for $|\omega_1|\rightarrow 0$, to find an expansion of the form $$ \wp(z,\omega_1,\omega_2) = \sum_{n=-N}^{\infty} F_n(z,\omega_1) B_n(\omega_1) $$ and similar for $\zeta$, where I would be happy with the basis of functions being either $B_n( \omega_1) = \omega_1^n$ or $B_n(\omega_1) = \exp(-n/\omega_1)$ or possibly a combination of the two. Here, $N$ is an integer and the $F_n$ are the coefficients.

I have tried finding this online and tried to use several trigonometric expansions to get this form, but failed until now. In particular, I used the Digital library of mathematical functions:


Is there anything known about expansions of this type?

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    $\begingroup$ I would suggest to first express $\wp$ and $\zeta$ in terms of theta functions using these formulas. Your asymptotics corresponds to taking $\tau \to 0$ (or $q\to 1$). It can be easily treated by applying Jacobi imaginary transformation, so that the transformed parameters behave as $\tau'\to i\infty$ (or $q'\to 0$). Then your expansion corresponds to series representation of the transformed theta functions. $\endgroup$ Aug 21 '15 at 11:04
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    $\begingroup$ @L.G. Thanks for your suggestion! I think I have succesfully used your suggestion to produce a series for $\wp$ and for $\zeta$. I will try to write an answer when I have more time. $\endgroup$
    – Hrodelbert
    Aug 24 '15 at 11:57

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