# §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 $\wp\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-\frac{1}{3}+{\csc}^{2}% \left(\frac{\pi z}{2\omega_{1}}\right)+8\left(1-\cos\left(\frac{\pi z}{\omega_% {1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),$
 23.12.2 $\zeta\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(\frac{z}{3}+\frac{2% \omega_{1}}{\pi}\cot\left(\frac{\pi z}{2\omega_{1}}\right)-8\left(z-\frac{% \omega_{1}}{\pi}\sin\left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+O\left(q% ^{4}\right)\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$: cotangent function, $\sin\NVar{z}$: sine function, $\mathbb{L}$: lattice, $q$: nome, $z$: complex and $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators Referenced by: §23.12, Erratum (V1.0.23) for Equation (23.12.2) Permalink: http://dlmf.nist.gov/23.12.E2 Encodings: TeX, pMML, png Correction (effective with 1.0.23): The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted. Suggested 2019-05-27 by Blagoje Oblak See also: Annotations for §23.12 and Ch.23
 23.12.3 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\pi^{2}z^{2}}{24% \omega_{1}^{2}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-% \left(\frac{\pi^{2}z^{2}}{\omega_{1}^{2}}-4{\sin}^{2}\left(\frac{\pi z}{2% \omega_{1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),$

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

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

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