§23.12 Asymptotic Approximations

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Keywords:: Weierstrass elliptic functions, asymptotic approximations
Notes:: These approximations follow from the expansions given in §23.8(ii). For (23.12.4) use Lawden (1989, (6.2.7)) and (20.4.8).
Permalink:: http://dlmf.nist.gov/23.12
See also:: Annotations for Ch.23

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : 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
See also:: Annotations for §23.12 and Ch.23

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\sin\NVar{z}$ : 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
See also:: Annotations for §23.12 and Ch.23

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, : 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).