§23.12 Asymptotic Approximations

If $q\>(=e^{\pi i\omega_{3}/\omega_{1}})\to 0$ with $\omega_{1}$ and $z$ fixed, then

 23.12.1 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-% \frac{1}{3}+{\mathop{\csc\/}\nolimits^{2}}\!\left(\frac{\pi z}{2\omega_{1}}% \right)+8\left(1-\mathop{\cos\/}\nolimits\!\left(\frac{\pi z}{\omega_{1}}% \right)\right)q^{2}+\mathop{O\/}\nolimits\!\left(q^{4}\right)\right),$
 23.12.2 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left% (\frac{z}{3}+\frac{2\omega_{1}}{\pi}\mathop{\cot\/}\nolimits\!\left(\frac{\pi z% }{\omega_{1}}\right)-8\left(z-\frac{\omega_{1}}{\pi}\mathop{\sin\/}\nolimits\!% \left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+\mathop{O\/}\nolimits\!\left% (q^{4}\right)\right),$
 23.12.3 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\pi^{2}z^{2}}{24\omega_{1}^{2}}\right)\mathop{\sin\/% }\nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-\left(\frac{\pi^{2% }z^{2}}{\omega_{1}^{2}}-4{\mathop{\sin\/}\nolimits^{2}}\!\left(\frac{\pi z}{2% \omega_{1}}\right)\right)q^{2}+\mathop{O\/}\nolimits\!\left(q^{4}\right)\right),$

provided that $z\notin\mathbb{L}$ in the case of (23.12.1) and (23.12.2). Also,

 23.12.4 $\eta_{1}=\frac{\pi^{2}}{4\omega_{1}}\left(\frac{1}{3}-8q^{2}+\mathop{O\/}% \nolimits\!\left(q^{4}\right)\right),$

with similar results for $\eta_{2}$ and $\eta_{3}$ obtainable by use of (23.2.14).