# nome

(0.000 seconds)

## 1—10 of 25 matching pages

##### 1: 22.2 Definitions
###### §22.2 Definitions
The nome $q$ is given in terms of the modulus $k$ by
$k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q\right)},$
$k^{\prime}=\frac{{\theta_{4}^{2}}\left(0,q\right)}{{\theta_{3}^{2}}\left(0,q% \right)},$
##### 2: 22.11 Fourier and Hyperbolic Series
Throughout this section $q$ and $\zeta$ are defined as in §22.2. If $q\exp\left(2|\Im\zeta|\right)<1$, then
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
Next, if $q\exp\left(|\Im\zeta|\right)<1$, then … Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\exp\left(2|\Im\zeta|\right)<1$, …
##### 3: 20.4 Values at $z$ = 0
20.4.2 $\theta_{1}'\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)^{3},$
20.4.3 $\theta_{2}\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)\left(1+q^{2n}\right)^{2},$
20.4.4 $\theta_{3}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+q^{2n-1}\right)^{2},$
20.4.5 $\theta_{4}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-q^{2n-1}\right)^{2}.$
20.4.6 $\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).$
##### 4: 19.5 Maclaurin and Related Expansions
For Jacobi’s nome $q$:
19.5.5 $q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)=r+8r^{2}+84r% ^{3}+992r^{4}+\cdots,$ $r=\frac{1}{16}k^{2}$, $0\leq k\leq 1$.
19.5.6 $q=\lambda+2\lambda^{5}+15\lambda^{9}+150\lambda^{13}+1707\lambda^{17}+\cdots,$ $0\leq k\leq 1$,
##### 6: 20.1 Special Notation
 $m$, $n$ integers. … 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. $e^{i\alpha\pi\tau}$ for $\alpha\in\mathbb{R}$ (resolving issues of choice of branch). …
##### 7: 20.8 Watson’s Expansions
20.8.1 $\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.$
##### 8: 22.1 Special Notation
 $x,y$ real variables. … nome. $0\leq q<1$ except in §22.17; see also §20.1. …
##### 9: 23.17 Elementary Properties
23.17.5 $1728J\left(\tau\right)=q^{-2}+744+1\;96884q^{2}+214\;93760q^{4}+\cdots,$
23.17.7 $\lambda\left(\tau\right)=16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-% 1}}\right)^{8},$
##### 10: 23.15 Definitions
In §§23.1523.19, $k$ and $k^{\prime}$ $(\in\mathbb{C})$ denote the Jacobi modulus and complementary modulus, respectively, and $q=e^{i\pi\tau}$ ($\Im\tau>0$) denotes the nome; compare §§20.1 and 22.1. …
23.15.1 $q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}{K\left(k\right)}\right),$
23.15.6 $\lambda\left(\tau\right)=\frac{{\theta_{2}^{4}}\left(0,q\right)}{{\theta_{3}^{% 4}}\left(0,q\right)};$
23.15.7 $J\left(\tau\right)=\frac{\left({\theta_{2}^{8}}\left(0,q\right)+{\theta_{3}^{8% }}\left(0,q\right)+{\theta_{4}^{8}}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},$
23.15.8 $\theta_{1}'\left(0,q\right)=\ifrac{\partial\theta_{1}\left(z,q\right)}{% \partial z}|_{z=0}.$