Jacobi Elliptic Functions
Let $u$ and $k$ be real numbers with $k \geqslant 0$, the Jacobi elliptic functions $sn(u,k),cn(u,k), dn(u,k)$ are defined as follows, given \[u=\int_0^\phi \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}\] and three functions \[ \begin{array}{rcl} sn(u,k) & = & \sin\phi \\ cn(u,k) & = & \cos\phi \\ dn(u,k) & = & \sqrt{1-k^2\sin^2\phi} \end{array} \] where $\phi$ is called the amplitude and $k$ the elliptic modulus.
This calculator evaluates three functions above and the following nine functions.
- Jacobi elliptic function $cd$ \[ cd(u,k)=\frac{cn(u,k)}{dn(u,k)}\]
- Jacobi elliptic function $cs$ \[ cs(u,k)=\frac{cn(u,k)}{sn(u,k)}\]
- Jacobi elliptic function $dc$ \[ dc(u,k)=\frac{dn(u,k)}{cn(u,k)}\]
- Jacobi elliptic function $ds$ \[ ds(u,k)=\frac{dn(u,k)}{sn(u,k)}\]
- Jacobi elliptic function $nc$ \[ nc(u,k)=\frac{1}{cn(u,k)}\]
- Jacobi elliptic function $nd$ \[ nd(u,k)=\frac{1}{dn(u,k)}\]
- Jacobi elliptic function $ns$ \[ ns(u,k)=\frac{1}{sn(u,k)}\]
- Jacobi elliptic function $sc$ \[ sc(u,k)=\frac{sn(u,k)}{cn(u,k)}\]
- Jacobi elliptic function $sd$ \[ sd(u,k)=\frac{sn(u,k)}{dn(u,k)}\]