# Weierstrass product

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## 7 matching pages

##### 2: 23.8 Trigonometric Series and Products
###### §23.8(iii) Infinite Products
23.8.6 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\eta_{1}z^{2}}{2% \omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\prod_{n=1}^{% \infty}\frac{1-2q^{2n}\cos\left(\pi z/\omega_{1}\right)+q^{4n}}{(1-q^{2n})^{2}},$
23.8.7 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\eta_{1}z^{2}}{2% \omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\prod_{n=1}^{% \infty}\frac{\sin\left(\pi(2n\omega_{3}+z)/(2\omega_{1})\right)\sin\left(\pi(2% n\omega_{3}-z)/(2\omega_{1})\right)}{{\sin}^{2}\left(\pi n\omega_{3}/\omega_{1% }\right)}.$
##### 3: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … nome. discriminant ${g_{2}}^{3}-27{g_{3}}^{2}$. … Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$.
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 4: 23.2 Definitions and Periodic Properties
23.2.6 $\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-\frac{% z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$
##### 5: 23.10 Addition Theorems and Other Identities
23.10.13 $\sigma\left(nz\right)=A_{n}e^{-n(n-1)(\eta_{1}+\eta_{3})z}\prod_{j=0}^{n-1}% \prod_{\ell=0}^{n-1}\sigma\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega% _{3}\right),$
23.10.14 $A_{n}=n\prod_{j=0}^{n-1}\prod_{\begin{subarray}{c}\ell=0\\ \ell\neq j\end{subarray}}^{n-1}\frac{1}{\sigma\left((2j\omega_{1}+2\ell\omega_% {3})/n\right)}.$
##### 7: Errata
• Chapter 19

Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

• 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).

• References

Some references were added to §§7.25(ii), 7.25(iii), 7.25(vi), 8.28(ii), and to ¶Products (in §10.74(vii)) and §10.77(ix).

• 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.