§23.12 Asymptotic Approximations

Notes:: These approximations follow from the expansions given in §23.8(ii). For (23.12.4) use Lawden (1989, Eq. 6.2.7) and (20.4.8).
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.12

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

23.12.1 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-% \frac{1}{3}+{\mathop{\csc\/}\nolimits^{{2}}}\!\left(\frac{\pi z}{2\omega_{1}}% \right)+8\left(1-\mathop{\cos\/}\nolimits\!\left(\frac{\pi z}{\omega_{1}}% \right)\right)q^{2}+\mathop{O\/}\nolimits\!\left(q^{4}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathbb{L}$ : lattice, : nome, : 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

23.12.2 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left% (\frac{z}{3}+\frac{2\omega_{1}}{\pi}\mathop{\cot\/}\nolimits\!\left(\frac{\pi z% }{\omega_{1}}\right)-8\left(z-\frac{\omega_{1}}{\pi}\mathop{\sin\/}\nolimits\!% \left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+\mathop{O\/}\nolimits\!\left% (q^{4}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathbb{L}$ : lattice, : nome, : 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.E2
Encodings:: TeX, pMML, png

23.12.3 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\pi^{2}z^{2}}{24\omega_{1}^{2}}\right)\mathop{\sin\/% }\nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-\left(\frac{\pi^{2% }z^{2}}{\omega_{1}^{2}}-4{\mathop{\sin\/}\nolimits^{{2}}}\!\left(\frac{\pi z}{% 2\omega_{1}}\right)\right)q^{2}+\mathop{O\/}\nolimits\!\left(q^{4}\right)% \right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathbb{L}$ : lattice, : nome, : 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

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : nome, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/23.12.E4
Encodings:: TeX, pMML, png

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