§23.12 Asymptotic Approximations

If $q(={\mathrm{e}}^{\pi \mathrm{i}{\omega }_3/{\omega }_1})\to 0$ with ${\omega }_1$ and $z$ fixed, then

23.12.1		$$\mathrm{\wp }\left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(-\frac{1}{3}+{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)+8\left(1-\cos \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators Referenced by: §23.12 Permalink: http://dlmf.nist.gov/23.12.E1 Encodings: TeX, pMML, png See also: Annotations for §23.12 and Ch.23

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: $O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: 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		$$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators Permalink: http://dlmf.nist.gov/23.12.E3 Encodings: TeX, pMML, png See also: Annotations for §23.12 and Ch.23

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

23.12.4		$${\eta }_1=\frac{{\pi }^2}{4{\omega }_1}\left(\frac{1}{3}-8q^2+O\left(q^4\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number Referenced by: §23.12 Permalink: http://dlmf.nist.gov/23.12.E4 Encodings: TeX, pMML, png See also: Annotations for §23.12 and Ch.23

with similar results for ${\eta }_2$ and ${\eta }_3$ obtainable by use of (23.2.14).