# Problem with EllipticE documentation

The complete elliptic integral of the second kind, EllipticE, is defined as,

Integrate[Sqrt[1-m Sin[t]^2],{t,0,z}]


According to the version 8 docs, the first of the "Possible Issues" is supposed to evaluate Integrate[Sqrt[1-m Sin[t]^2],{t,0,z}] as

If[(m Sin[z]^2 \[NotElement] Reals ||
Re[m Sin[z]^2] <= 1) && (Csc[z]^2/m \[NotElement] Reals ||
Re[Csc[z]^2/m] <= 0 || Re[Csc[z]^2/m] >= 1) &&
2 + m Cos[2 z] != m,
EllipticE[z, m],
Integrate[Sqrt[1 - m Sin[t]^2], {t, 0, z}, Assumptions ->
2 + m Cos[2 z] == m || (((2 - m + m Cos[2 z]) Csc[z]^2)/m \[Element] Reals && -2 <
Re[((2 - m + m Cos[2 z]) Csc[z]^2)/m] < 0) || (Re[m Sin[z]^2] >
1 && m Sin[z]^2 \[Element] Reals)]]


whereas I get simply

EllipticE[z,m]


Is this a bug in the docs?

• Same here on 8.0.4/Win7-64 – Sjoerd C. de Vries Jan 30 '12 at 20:36
• You can always submit any issues you find with the documentation to support@wolfram.com. – Searke Jan 31 '12 at 21:24
• @Searke Done. :) – JxB Jan 31 '12 at 22:13

I just tried it with both Mathematica 7 and 8, and Mathematica 7 gives the result from the documentation, while Mathematica 8 indeed gives just EllipticE[z, m].
Therefore I conclude Wolfram modified Integrate but forgot to update this piece of documentation.
• @SjoerdC.deVries: I admit that I didn't read the documentation, but relied on the quote above. Since it was referred to as "Issue" I assumed ― wrongly, as it seems ― that it was a limitation of Integrate. OTOH it's not the only place where Mathematica makes implicit assumtions on integration: Integrate[x^n,x] gives x^(1 + n)/(1 + n), without any special case for n=-1. I now changed "improved" to "modified". – celtschk Jan 30 '12 at 21:06