# lattice parameter

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## 1—10 of 19 matching pages

##### 1: 20.2 Definitions and Periodic Properties
20.2.2 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(z,q\right)=2\sum\limits_{% n=0}^{\infty}q^{(n+\frac{1}{2})^{2}}\cos\left((2n+1)z\right),$
##### 2: 20.6 Power Series
20.6.2 $\theta_{1}\left(\pi z\middle|\tau\right)=\pi z\theta_{1}'\left(0\middle|\tau% \right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),$
20.6.3 $\theta_{2}\left(\pi z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)% \exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}\right),$
20.6.7 $\alpha_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{% 1}{2}+n\tau)^{-2j},$
20.6.8 $\beta_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1% }{2}+(n-\tfrac{1}{2})\tau)^{-2j},$
20.6.9 $\gamma_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m+(n-% \tfrac{1}{2})\tau)^{-2j},$
##### 3: 20.7 Identities
20.7.15 $A\equiv A(\tau)=\ifrac{1}{\theta_{4}\left(0\middle|2\tau\right)},$
20.7.16 $\theta_{1}\left(2z\middle|2\tau\right)=A\theta_{1}\left(z\middle|\tau\right)% \theta_{2}\left(z\middle|\tau\right),$
20.7.19 $\theta_{4}\left(2z\middle|2\tau\right)=A\theta_{3}\left(z\middle|\tau\right)% \theta_{4}\left(z\middle|\tau\right).$
###### §20.7(viii) Transformations of LatticeParameter
20.7.28 $\theta_{3}\left(z\middle|\tau+1\right)=\theta_{4}\left(z\middle|\tau\right),$
##### 5: 20.1 Special Notation
 $m$, $n$ integers. … the lattice parameter, $\Im\tau>0$. the nome, $q=e^{i\pi\tau}$, $0<\left|q\right|<1$. Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$, $\Im\tau>0$, so that $0 and $\tau$ and $q$ are uniquely related. …
##### 6: 20.5 Infinite Products and Related Results
20.5.5 $\theta_{1}\left(z\middle|\tau\right)=\theta_{1}'\left(0\middle|\tau\right)\sin z% \prod_{n=1}^{\infty}\frac{\sin\left(n\pi\tau+z\right)\sin\left(n\pi\tau-z% \right)}{{\sin}^{2}\left(n\pi\tau\right)},$
20.5.6 $\theta_{2}\left(z\middle|\tau\right)=\theta_{2}\left(0\middle|\tau\right)\cos z% \prod_{n=1}^{\infty}\frac{\cos\left(n\pi\tau+z\right)\cos\left(n\pi\tau-z% \right)}{{\cos}^{2}\left(n\pi\tau\right)},$
20.5.7 $\theta_{3}\left(z\middle|\tau\right)=\theta_{3}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})\pi\tau+z\right)\cos\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\cos}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)},$
20.5.8 $\theta_{4}\left(z\middle|\tau\right)=\theta_{4}\left(0\middle|\tau\right)\prod% _{n=1}^{\infty}\frac{\sin\left((n-\tfrac{1}{2})\pi\tau+z\right)\sin\left((n-% \tfrac{1}{2})\pi\tau-z\right)}{{\sin}^{2}\left((n-\tfrac{1}{2})\pi\tau\right)}.$
20.5.9 $\theta_{3}\left(\pi z\middle|\tau\right)=\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{% 2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right),$
##### 7: 20.9 Relations to Other Functions
20.9.1 $k={\theta_{2}}^{2}\left(0\middle|\tau\right)/{\theta_{3}}^{2}\left(0\middle|% \tau\right)$
##### 8: 20.11 Generalizations and Analogs
20.11.5 ${{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)={\theta_{3}}^{2}% \left(0\middle|\tau\right);$
##### 9: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … lattice parameter ($\Im\tau>0$). …
##### 10: 20.13 Physical Applications
20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$