# sigma function

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##### 1: 32.6 Hamiltonian Structure
The function
32.6.7 $q=-\sigma^{\prime},$
32.6.8 $p=-\sigma^{\prime\prime},$
The function $\sigma(z)=\mathrm{H}_{\mbox{\scriptsize II}}(q,p,z)$ defined by (32.6.9) satisfies … The function
##### 2: 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 $\eta\left(\tau\right)$. …
##### 3: 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).
##### 4: 10.73 Physical Applications
For applications of the Rayleigh function $\sigma_{n}\left(\nu\right)$10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).
##### 5: 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 $\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).$
##### 6: 33.13 Complex Variable and Parameters
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 7: 26.2 Basic Definitions
The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. …
##### 8: 10.21 Zeros
If $\sigma_{\nu}$ is a zero of $\mathscr{C}_{\nu}'\left(z\right)$, then … The parameter $t$ may be regarded as a continuous variable and $\rho_{\nu}$, $\sigma_{\nu}$ as functions $\rho_{\nu}(t)$, $\sigma_{\nu}(t)$ of $t$. … The functions $\rho_{\nu}(t)$ and $\sigma_{\nu}(t)$ are related to the inverses of the phase functions $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ defined in §10.18(i): if $\nu\geq 0$, then … The Rayleigh function $\sigma_{n}\left(\nu\right)$ is defined by
10.21.55 $\sigma_{n}\left(\nu\right)=\sum_{m=1}^{\infty}(j_{\nu,m})^{-2n},$ $n=1,2,3,\dots$.
##### 9: 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 $\sigma\left(z\right)$. …
##### 10: 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 $\sigma\left(\omega_{1}\right)=2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}% \omega_{1}\right)\theta_{2}\left(0,q\right)}{\pi\theta_{1}'\left(0,q\right)},$
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). … For representations of general elliptic functions23.2(iii)) in terms of $\sigma\left(z\right)$ and $\wp\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta\left(z\right)$ see Lawden (1989, §8.11). …