# Jacobi nome

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## 1—10 of 19 matching pages

##### 1: 19.5 Maclaurin and Related Expansions
For Jacobi’s nome $q$:
19.5.5 $q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)=r+8r^{2}+84r% ^{3}+992r^{4}+\cdots,$ $r=\frac{1}{16}k^{2}$, $0\leq k\leq 1$.
##### 2: 22.2 Definitions
$k=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},$
$k^{\prime}=\frac{{\theta_{4}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},$
22.2.4 $\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},$
22.2.7 $\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}}^{2}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},$
##### 3: 22.16 Related Functions
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 $\xi=x/{\theta_{3}}^{2}\left(0,q\right)$. …
##### 4: 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. …
##### 5: 20.4 Values at $z$ = 0
20.4.1 $\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,$
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 $\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).$
##### 6: 22.11 Fourier and Hyperbolic Series
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.2 $\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
22.11.3 $\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$
22.11.4 $\operatorname{cd}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)% ^{n}q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.13 ${\operatorname{sn}}^{2}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}% \right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}% \cos\left(2n\zeta\right).$
##### 7: 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 ${\theta_{3}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{2}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{3}}^{2}\left(z,q\right),$
20.7.3 ${\theta_{2}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{3}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right),$
20.7.4 ${\theta_{2}}^{2}\left(0,q\right){\theta_{3}}^{2}\left(z,q\right)={\theta_{4}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{3}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right).$
20.7.5 ${\theta_{3}}^{4}\left(0,q\right)={\theta_{2}}^{4}\left(0,q\right)+{\theta_{4}}% ^{4}\left(0,q\right).$
##### 9: 20.8 Watson’s Expansions
20.8.1 $\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.$
##### 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. …
23.15.6 $\lambda\left(\tau\right)=\frac{{\theta_{2}}^{4}\left(0,q\right)}{{\theta_{3}}^% {4}\left(0,q\right)};$
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 $\theta_{1}'\left(0,q\right)=\ifrac{\partial\theta_{1}\left(z,q\right)}{% \partial z}|_{z=0}.$
23.15.9 $\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/% 3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right).$