# Weierstrass sigma function

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##### 1: 23.1 Special Notation
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 $\eta\left(\tau\right)$. …
##### 2: 23.2 Definitions and Periodic Properties
###### §23.2(ii) Weierstrass Elliptic Functions
23.2.6 ${}\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-% \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$
The function $\sigma\left(z\right)$ is entire and odd, with simple zeros at the lattice points. … For $j=1,2,3$, the function $\sigma\left(z\right)$ satisfies … For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).
##### 3: 23.10 Addition Theorems and Other Identities
23.10.3 $\frac{\sigma\left(u+v\right)\sigma\left(u-v\right)}{{\sigma}^{2}\left(u\right)% {\sigma}^{2}\left(v\right)}=\wp\left(v\right)-\wp\left(u\right),$
23.10.4 $\sigma\left(u+v\right)\sigma\left(u-v\right)\sigma\left(x+y\right)\sigma\left(% x-y\right)+\sigma\left(v+x\right)\sigma\left(v-x\right)\sigma\left(u+y\right)% \sigma\left(u-y\right)+{\sigma\left(x+u\right)\sigma\left(x-u\right)\sigma% \left(v+y\right)\sigma\left(v-y\right)=0.}$
For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). …
23.10.10 $\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$
23.10.19 $\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).$
##### 4: 23.6 Relations to Other Functions
23.6.9 $\sigma\left(z\right)=2\omega_{1}\exp\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}% \right)\frac{\theta_{1}\left(\pi z/(2\omega_{1}),q\right)}{\pi\theta_{1}'\left% (0,q\right)},$
23.6.10 $\sigma\left(\omega_{1}\right)=2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}% \omega_{1}\right)\theta_{2}\left(0,q\right)}{\pi\theta_{1}'\left(0,q\right)},$
23.6.15 $\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left(\omega_{j}\right)}=\exp% \left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)\frac{\theta_{j+1}% \left(z,q\right)}{\theta_{j+1}\left(0,q\right)},$ $j=1,2,3$.
For further results for the $\sigma$-function see Lawden (1989, §6.2). … For representations of the Jacobi functions $\operatorname{sn}$, $\operatorname{cn}$, and $\operatorname{dn}$ as quotients of $\sigma$-functions see Lawden (1989, §§6.2, 6.3). …
##### 5: 23.3 Differential Equations
As functions of $g_{2}$ and $g_{3}$, $\wp\left(z;g_{2},g_{3}\right)$ and $\zeta\left(z;g_{2},g_{3}\right)$ are meromorphic and $\sigma\left(z;g_{2},g_{3}\right)$ is entire. …
##### 6: 23.4 Graphics
Line graphs of the Weierstrass functions $\wp\left(x\right)$, $\zeta\left(x\right)$, and $\sigma\left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … Surfaces for the Weierstrass functions $\wp\left(z\right)$, $\zeta\left(z\right)$, and $\sigma\left(z\right)$. …
##### 7: 23.9 Laurent and Other Power Series
23.9.7 $\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},$
##### 8: 23.12 Asymptotic Approximations
23.12.3 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\pi^{2}z^{2}}{24% \omega_{1}^{2}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-% \left(\frac{\pi^{2}z^{2}}{\omega_{1}^{2}}-4{\sin}^{2}\left(\frac{\pi z}{2% \omega_{1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),$
##### 9: 23.8 Trigonometric Series and 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 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\eta_{1}z^{2}}{2% \omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\prod_{n=1}^{% \infty}\frac{\sin\left(\pi(2n\omega_{3}+z)/(2\omega_{1})\right)\sin\left(\pi(2% n\omega_{3}-z)/(2\omega_{1})\right)}{{\sin}^{2}\left(\pi n\omega_{3}/\omega_{1% }\right)}.$
##### 10: 19.25 Relations to Other Functions
19.25.40 $z+2\omega=\pm\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),$