Difference between revisions of "Jacobi sn"

From specialfunctionswiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 4: Line 4:
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Domcoljacobisn.png|[[Domain coloring]] of $\mathrm{sn}$ corresponding to $m=0.8$.
+
File:Complexjacobisn,m=0.8plot.png|[[Domain coloring]] of $\mathrm{sn}$ with $m=0.8$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
Line 18: Line 18:
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf Special functions by Leon Hall]
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf Special functions by Leon Hall]
  
<center>{{:Jacobi elliptic functions footer}}</center>
+
{{:Jacobi elliptic functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[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