Difference between revisions of "Jacobi cn"

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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
 
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
 
$$\mathrm{cn \hspace{2pt}} u = \cos \phi = \sqrt{1-x^2}.$$
 
$$\mathrm{cn \hspace{2pt}} u = \cos \phi = \sqrt{1-x^2}.$$
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<div align="center">
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<gallery>
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File:Complexjacobicn,m=0.8plot.png|[[Domain coloring]] of $\mathrm{cn}$ with $m=0.8$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
 
#$\mathrm{sn \hspace{2pt}}^2u+\mathrm{cn \hspace{2pt}}^2u=1$
 
#$\mathrm{sn \hspace{2pt}}^2u+\mathrm{cn \hspace{2pt}}^2u=1$
 
#$\mathrm{cn \hspace{2pt}}(0)=1$
 
#$\mathrm{cn \hspace{2pt}}(0)=1$
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#$\mathrm{cn \hspace{2pt}}$ is an even function
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#$\dfrac{d}{du}\mathrm{sn \hspace{2pt}} u =\mathrm{cn \hspace{2pt}}(u)\mathrm{dn \hspace{2pt}}(u)$
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=References=
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[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf Special functions by Leon Hall]
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{{:Jacobi elliptic functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 19:06, 5 July 2016

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