# Weierstrass form

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## 9 matching pages

##### 2: 23.21 Physical Applications
The Weierstrass function $\wp$ plays a similar role for cubic potentials in canonical form $g_{3}+g_{2}x-4x^{3}$. … Another form is obtained by identifying $e_{1}$, $e_{2}$, $e_{3}$ as lattice roots (§23.3(i)), and setting …
we have …
##### 4: 23.9 Laurent and Other Power Series
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\wp\left(z\right)\to 0$. …
##### 6: Errata
• Equation (23.6.15)
23.6.15 $\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left(\omega_{j}\right)}=\exp% \left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)\frac{\theta_{j+1}% \left(z,q\right)}{\theta_{j+1}\left(0,q\right)},$ $j=1,2,3$

The factor $\exp\left(\eta_{j}u+\frac{\eta_{j}u^{2}}{2\omega_{1}}\right)$ has been corrected to be $\exp\left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)$.

Reported by Jan Felipe van Diejen on 2021-02-10

• Subsection 19.25(vi)

This subsection has been significantly updated. In particular, the following formulae have been corrected. Equation (19.25.35) has been replaced by

19.25.35 $z+2\omega=\pm R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp% \left(z\right)-e_{3}\right),$

in which the left-hand side $z$ has been replaced by $z+2\omega$ for some $2\omega\in\mathbb{L}$, and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.37) has been replaced by

19.25.37 $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)=\pm 2R_{G}\left(\wp% \left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right),$

in which the left-hand side $\zeta\left(z\right)+z\wp\left(z\right)$ has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$ and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.39) has been replaced by

19.25.39 $\zeta\left(\omega_{j}\right)+\omega_{j}e_{j}=2R_{G}\left(0,e_{j}-e_{k},e_{j}-e% _{\ell}\right),$

in which the left-hand side $\eta_{j}$ was replaced by $\zeta\left(\omega_{j}\right)$, for some $2\omega_{j}\in\mathbb{L}$ and $\wp\left(\omega_{j}\right)=e_{j}$. Equation (19.25.40) has been replaced by

19.25.40 $z+2\omega=\pm\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),$

in which the left-hand side $z$ has been replaced by $z+2\omega$, and the right-hand side was multiplied by $\pm 1$. For more details see §19.25(vi).

• Subsection 19.25(vi)

The Weierstrass lattice roots $e_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$, and lattice invariants $g_{2}$, $g_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

• Equation (19.25.37)

The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

• Equation (23.2.4)
23.2.4 $\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right)$

Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$.

Reported 2012-02-16 by James D. Walker.

• ##### 7: Software Index
 Open Source With Book Commercial … 23 Weierstrass Elliptic and Modular Functions …
• Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

• ##### 8: 1.9 Calculus of a Complex Variable
or in polar form ((1.9.3)) $u$ and $v$ satisfy … or its limiting form, and is invariant under bilinear transformations. …
##### 9: 1.10 Functions of a Complex Variable
A cut neighborhood is formed by deleting a ray emanating from the center. … It should be noted that different branches of $(w-w_{0})^{1/\mu}$ used in forming $(w-w_{0})^{n/\mu}$ in (1.10.16) give rise to different solutions of (1.10.12). …