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## 11—20 of 25 matching pages

##### 11: 20.7 Identities
20.7.1 ${\theta_{3}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)={\theta_{4}^{% 2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)+{\theta_{2}^{2}}\left(0,q% \right){\theta_{2}^{2}}\left(z,q\right),$
20.7.2 ${\theta_{3}^{2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)={\theta_{2}^{% 2}}\left(0,q\right){\theta_{1}^{2}}\left(z,q\right)+{\theta_{4}^{2}}\left(0,q% \right){\theta_{3}^{2}}\left(z,q\right),$
20.7.5 ${\theta_{3}^{4}}\left(0,q\right)={\theta_{2}^{4}}\left(0,q\right)+{\theta_{4}^% {4}}\left(0,q\right).$
##### 12: 20.5 Infinite Products and Related Results
20.5.2 $\theta_{2}\left(z,q\right)=2q^{1/4}\cos z\prod\limits_{n=1}^{\infty}{\left(1-q% ^{2n}\right)}{\left(1+2q^{2n}\cos\left(2z\right)+q^{4n}\right)},$
20.5.3 $\theta_{3}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right),$
20.5.4 $\theta_{4}\left(z,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-2q^{2n-1}\cos\left(2z\right)+q^{4n-2}\right).$
20.5.9 $\theta_{3}\left(\pi z\middle|\tau\right)=\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{% 2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right),$
##### 13: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … nome. …
##### 14: 23.12 Asymptotic Approximations
23.12.1 $\wp\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-\frac{1}{3}+{\csc^{2}}% \left(\frac{\pi z}{2\omega_{1}}\right)+8\left(1-\cos\left(\frac{\pi z}{\omega_% {1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),$
23.12.2 $\zeta\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(\frac{z}{3}+\frac{2% \omega_{1}}{\pi}\cot\left(\frac{\pi z}{2\omega_{1}}\right)-8\left(z-\frac{% \omega_{1}}{\pi}\sin\left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+O\left(q% ^{4}\right)\right),$
23.12.3 $\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\pi^{2}z^{2}}{24% \omega_{1}^{2}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-% \left(\frac{\pi^{2}z^{2}}{\omega_{1}^{2}}-4{\sin^{2}}\left(\frac{\pi z}{2% \omega_{1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),$
23.12.4 $\eta_{1}=\frac{\pi^{2}}{4\omega_{1}}\left(\frac{1}{3}-8q^{2}+O\left(q^{4}% \right)\right),$
##### 15: 20.2 Definitions and Periodic Properties
20.2.2 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(z,q\right)=2\sum\limits_{% n=0}^{\infty}q^{(n+\frac{1}{2})^{2}}\cos\left((2n+1)z\right),$
##### 16: 20.11 Generalizations and Analogs
20.11.6 $\varphi_{m,1}\left(z,q\right)=\frac{\theta_{1}'\left(0,q\right)\theta_{m}\left% (z,q\right)}{\theta_{m}\left(0,q\right)\theta_{1}\left(z,q\right)},$ $m=2,3,4$,
20.11.7 $\varphi_{1,n}\left(z,q\right)=\frac{\theta_{n}\left(0,q\right)\theta_{1}\left(% z,q\right)}{\theta_{1}'\left(0,q\right)\theta_{n}\left(z,q\right)},$ $n=2,3,4$,
20.11.8 $\varphi_{m,n}\left(z,q\right)=\frac{\theta_{n}\left(0,q\right)\theta_{m}\left(% z,q\right)}{\theta_{m}\left(0,q\right)\theta_{n}\left(z,q\right)},$ $m,n=2,3,4$.
20.11.9 $\varphi_{m,n}\left(z,q\right)=\varphi_{m,1}\left(z,q\right)\varphi_{1,n}\left(% z,q\right)=\frac{1}{\varphi_{n,m}\left(z,q\right)}=\frac{\varphi_{m,1}\left(z,% q\right)}{\varphi_{n,1}\left(z,q\right)}=\frac{\varphi_{1,n}\left(z,q\right)}{% \varphi_{1,m}\left(z,q\right)}.$
##### 17: 22.21 Tables
Curtis (1964b) tabulates $\operatorname{sn}\left(mK/n,k\right)$, $\operatorname{cn}\left(mK/n,k\right)$, $\operatorname{dn}\left(mK/n,k\right)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. …
##### 18: 22.16 Related Functions
With $q$ as in (22.2.1) and $\zeta=\pi x/(2K)$,
22.16.9 $\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^% {n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}.$
22.16.30 $\mathcal{E}\left(x,k\right)=\frac{1}{{\theta_{3}^{2}}\left(0,q\right)\theta_{4% }\left(\xi,q\right)}\frac{\mathrm{d}}{\mathrm{d}\xi}\theta_{4}\left(\xi,q% \right)+\frac{E\left(k\right)}{K\left(k\right)}x,$
where $\xi=x/{\theta_{3}^{2}}\left(0,q\right)$. …
##### 19: 22.20 Methods of Computation
If either $\tau$ or $q=e^{i\pi\tau}$ is given, then we use $k={\theta_{2}^{2}}\left(0,q\right)/{\theta_{3}^{2}}\left(0,q\right)$, $k^{\prime}={\theta_{4}^{2}}\left(0,q\right)/{\theta_{3}^{2}}\left(0,q\right)$, $K=\frac{1}{2}\pi{\theta_{3}^{2}}\left(0,q\right)$, and $K^{\prime}=-i\tau K$, obtaining the values of the theta functions as in §20.14. …
##### 20: 20.9 Relations to Other Functions
20.9.3 $R_{F}\left(\frac{{\theta_{2}^{2}}\left(z,q\right)}{{\theta_{2}^{2}}\left(0,q% \right)},\frac{{\theta_{3}^{2}}\left(z,q\right)}{{\theta_{3}^{2}}\left(0,q% \right)},\frac{{\theta_{4}^{2}}\left(z,q\right)}{{\theta_{4}^{2}}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,$