Jacobi cn

Let $u=\displaystyle\int_0^x \dfrac{1}{\sqrt{(1-t^2)(1-mt^2)}}dt = \displaystyle\int_0^{\phi} \dfrac{1}{\sqrt{1-m\sin^2 \theta}} d\theta.$ Then we define $$\mathrm{cn \hspace{2pt}} u = \cos \phi = \sqrt{1-x^2}.$$

Properties

1. $\mathrm{sn \hspace{2pt}}^2u+\mathrm{cn \hspace{2pt}}^2u=1$
2. $\mathrm{cn \hspace{2pt}}(0)=1$
3. $\mathrm{cn \hspace{2pt}}$ is an even function
4. $\dfrac{d}{du}\mathrm{sn \hspace{2pt}} u =\mathrm{cn \hspace{2pt}}(u)\mathrm{dn \hspace{2pt}}(u)$

References

Special functions by Leon Hall

Jacobi Elliptic Functions

Jacobi $\mathrm{cn}$

Jacobi $\mathrm{cd}$

Jacobi $\mathrm{cs}$

Jacobi $\mathrm{dc}$

Jacobi $\mathrm{dn}$

Jacobi $\mathrm{ds}$

Jacobi $\mathrm{nc}$

Jacobi $\mathrm{nd}$

Jacobi $\mathrm{ns}$

Jacobi $\mathrm{sc}$

Jacobi $\mathrm{sd}$

Jacobi $\mathrm{sn}$