Difference between revisions of "Elliptic E"

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:Ellipticeplot.png|Plot of $E(m)$ on $[-10,1]$.
+
File:Ellipticeplot.png|Graph of $E$.
 
File:Complexellipticeplot.png|[[Domain coloring]] of $E$.
 
File:Complexellipticeplot.png|[[Domain coloring]] of $E$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
 +
 +
=Properties=
 +
[[E(m)=(pi/2)2F1(-1/2,1/2;1;m)]]<br />
  
 
=See Also=
 
=See Also=

Latest revision as of 04:54, 21 December 2017

If $m=k^2$ we define the complete elliptic integral of the second kind, $E$, to be $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} \mathrm{d}\theta.$$

Properties

E(m)=(pi/2)2F1(-1/2,1/2;1;m)

See Also

Elliptic K
Incomplete Elliptic E

References

"Special Functions" by Leon Hall