Difference between revisions of "Jacobi cn"
From specialfunctionswiki
(→Properties) |
(→Properties) |
||
| Line 6: | Line 6: | ||
#$\mathrm{cn \hspace{2pt}}(0)=1$ | #$\mathrm{cn \hspace{2pt}}(0)=1$ | ||
#$\mathrm{cn \hspace{2pt}}$ is an even function | #$\mathrm{cn \hspace{2pt}}$ is an even function | ||
| + | #$\dfrac{d}{du}\mathrm{sn \hspace{2pt}} u =\mathrm{cn \hspace{2pt}}(u)\mathrm{dn \hspace{2pt}}(u)$ | ||
Revision as of 07:32, 10 March 2015
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
- $\mathrm{sn \hspace{2pt}}^2u+\mathrm{cn \hspace{2pt}}^2u=1$
- $\mathrm{cn \hspace{2pt}}(0)=1$
- $\mathrm{cn \hspace{2pt}}$ is an even function
- $\dfrac{d}{du}\mathrm{sn \hspace{2pt}} u =\mathrm{cn \hspace{2pt}}(u)\mathrm{dn \hspace{2pt}}(u)$
