# Jacobi?s

(0.008 seconds)

## 1—10 of 38 matching pages

##### 1: 22.8 Addition Theorems
22.8.5 $\operatorname{sd}(u+v)=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname% {nd}v+\operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{% \prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
22.8.9 $\operatorname{sc}(u+v)=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname% {nc}v+\operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{% 2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$
22.8.10 $\operatorname{ns}(u+v)=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname% {cs}v-\operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},$
22.8.11 $\operatorname{ds}(u+v)=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname% {ns}v-\operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},$
22.8.12 $\operatorname{cs}(u+v)=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname% {ns}v-\operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u}.$
##### 2: 22.6 Elementary Identities
22.6.2 $1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),$
22.6.9 $\operatorname{sd}\left(2z,k\right)=\frac{2\operatorname{sd}\left(z,k\right)% \operatorname{cd}\left(z,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^% {\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},$
22.6.14 $\operatorname{ns}\left(2z,k\right)=\frac{{\operatorname{ns}}^{4}\left(z,k% \right)-k^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k% \right)\operatorname{ns}\left(z,k\right)},$
22.6.15 $\operatorname{ds}\left(2z,k\right)=\frac{k^{2}{k^{\prime}}^{2}+{\operatorname{% ds}}^{4}\left(z,k\right)}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}% \left(z,k\right)\operatorname{ns}\left(z,k\right)},$
22.6.16 $\operatorname{cs}\left(2z,k\right)=\frac{{\operatorname{cs}}^{4}\left(z,k% \right)-{k^{\prime}}^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}% \left(z,k\right)\operatorname{ns}\left(z,k\right)}.$
##### 3: 22.13 Derivatives and Differential Equations
22.13.5 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{% 2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)}{% \left(1+k^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)},$
22.13.9 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sc}\left(z,k\right)\right)^{% 2}=\left(1+{\operatorname{sc}}^{2}\left(z,k\right)\right)\left(1+{k^{\prime}}^% {2}{\operatorname{sc}}^{2}\left(z,k\right)\right),$
22.13.12 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cs}\left(z,k\right)\right)^{% 2}=\left(1+{\operatorname{cs}}^{2}\left(z,k\right)\right)\left({k^{\prime}}^{2% }+{\operatorname{cs}}^{2}\left(z,k\right)\right).$
22.13.17 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(z,k\right)=(k% ^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}{k^{\prime}}^{2}% {\operatorname{sd}}^{3}\left(z,k\right),$
22.13.21 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sc}\left(z,k\right)=(1% +{k^{\prime}}^{2})\operatorname{sc}\left(z,k\right)+2{k^{\prime}}^{2}{% \operatorname{sc}}^{3}\left(z,k\right),$
##### 4: 22.4 Periods, Poles, and Zeros
For example, the poles of $\operatorname{sn}\left(z,k\right)$, abbreviated as $\operatorname{sn}$ in the following tables, are at $z=2mK+(2n+1)iK^{\prime}$. … For example, $\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)$. …
##### 5: 27.9 Quadratic Characters
If an odd integer $P$ has prime factorization $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …
##### 6: 22.14 Integrals
22.14.4 $\int\operatorname{cd}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right),$
22.14.5 $\int\operatorname{sd}\left(x,k\right)\,\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),$
22.14.10 $\int\operatorname{ns}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ds}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
22.14.11 $\int\operatorname{ds}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
22.14.12 $\int\operatorname{cs}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{ds}\left(x,k\right)\right).$
##### 7: 22.5 Special Values
For example, at $z=K+iK^{\prime}$, $\operatorname{sn}\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$. … Table 22.5.2 gives $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ for other special values of $z$. For example, $\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}$. …
##### 8: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. … The notation $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\operatorname{sn}\left(z,k\right)$ are $\mathrm{sn}(z\mathpunct{|}m)$ and $\mathrm{sn}(z,m)$ with $m=k^{2}$; see Abramowitz and Stegun (1964) and Walker (1996). …
##### 9: 22.3 Graphics
Line graphs of the functions $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, $\operatorname{dn}\left(x,k\right)$, $\operatorname{cd}\left(x,k\right)$, $\operatorname{sd}\left(x,k\right)$, $\operatorname{nd}\left(x,k\right)$, $\operatorname{dc}\left(x,k\right)$, $\operatorname{nc}\left(x,k\right)$, $\operatorname{sc}\left(x,k\right)$, $\operatorname{ns}\left(x,k\right)$, $\operatorname{ds}\left(x,k\right)$, and $\operatorname{cs}\left(x,k\right)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, and $\operatorname{dn}\left(x,k\right)$ as functions of real arguments $x$ and $k$. …
##### 10: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … Jacobi symbol; see §27.9. …