Difference between revisions of "Jacobi cn"

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(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

  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. $\dfrac{d}{du}\mathrm{sn \hspace{2pt}} u =\mathrm{cn \hspace{2pt}}(u)\mathrm{dn \hspace{2pt}}(u)$