Difference between revisions of "Jacobi sn"

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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
 
$$\mathrm{sn \hspace{2pt}}u = \sin \phi = x.$$
 
$$\mathrm{sn \hspace{2pt}}u = \sin \phi = x.$$
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<div align="center">
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<gallery>
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File:Complexjacobisn,m=0.8plot.png|[[Domain coloring]] of $\mathrm{sn}$ with $m=0.8$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
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#$\mathrm{sn \hspace{2pt}}(0)=0$
 
#$\mathrm{sn \hspace{2pt}}(0)=0$
 
#$m \mathrm{sn \hspace{2pt}}^2 u + \mathrm{dn \hspace{2pt}}^2u=1$
 
#$m \mathrm{sn \hspace{2pt}}^2 u + \mathrm{dn \hspace{2pt}}^2u=1$
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#$\mathrm{sn \hspace{2pt}}$ is an odd 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 $$\mathrm{sn \hspace{2pt}}u = \sin \phi = x.$$

Properties

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

References

Special functions by Leon Hall

Jacobi Elliptic Functions