Editing Elliptic integral (section)

===Computation===

Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the [[arithmetic-geometric mean|arithmetic–geometric mean]].{{sfn|Carlson|2010|loc=19.8}}

Define sequences {{mvar|a<sub>n</sub>}} and {{mvar|g<sub>n</sub>}}, where {{math|1=''a''<sub>0</sub> = 1}}, {{math|1=''g''<sub>0</sub> = {{sqrt|1 − ''k''<sup>2</sup>}} = ''k''{{prime}}}} and the recurrence relations {{math|1=''a''<sub>''n'' + 1</sub> = {{sfrac|''a<sub>n</sub>'' + ''g<sub>n</sub>''|2}}}}, {{math|1=''g''<sub>''n'' + 1</sub> = {{sqrt|''a<sub>n</sub> g<sub>n</sub>''}}}} hold. Furthermore, define
<math display="block">c_n=\sqrt{\left|a_n^2-g_n^2\right|}.</math>

By definition,

<math display="block">a_\infty = \lim_{n\to\infty} a_n = \lim_{n\to\infty} g_n = \operatorname{agm}\left(1, \sqrt{1-k^2}\right).</math>

Also

<math display="block">\lim_{n\to\infty} c_n=0.</math>

Then

<math display="block">E(k) = \frac{\pi}{2a_\infty}\left(1-\sum_{n=0}^{\infty} 2^{n-1} c_n^2\right).</math>

In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all {{math|{{abs|''k''}} ≤ 1}}. To speed up computation further, the relation {{math|1=''c''<sub>''n'' + 1</sub> = {{sfrac|''c<sub>n</sub>''<sup>2</sup>|4''a''<sub>''n'' + 1</sub>}}}} can be used.

Furthermore, if {{math|1=''k''<sup>2</sup> = ''λ''(''i''{{sqrt|''r''}})}} and <math>r \isin \mathbb{Q}^+</math> (where {{mvar|λ}} is the [[modular lambda function]]), then {{math|''E''(''k'')}} is expressible in closed form in terms of
<math display="block">K(k)=\frac{\pi}{2\operatorname{agm}\left(1,\sqrt{1-k^2}\right)}</math>
and hence can be computed without the need for the infinite summation term. For example, {{math|1=''r'' = 1}}, {{math|1=''r'' = 3}} and {{math|1=''r'' = 7}} give, respectively,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 26, 161</ref>

<math display="block">E\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}K\left(\frac{1}{\sqrt{2}}\right)+\frac{\pi}{4K\left(\frac{1}{\sqrt{2}}\right)},</math>

and

<math display="block">E\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)=\frac{3+\sqrt{3}}{6}K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)+\frac{\pi\sqrt{3}}{12K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)},</math>

and

<math display="block">E\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)=\frac{7+2\sqrt{7}}{14}K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)+\frac{\pi\sqrt{7}}{28K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)}.</math>