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Re: Integrate modified in version 6?


On Jun 20, 4:38 am, dimitris <dimmec... at yahoo.com> wrote:
>
> Integrate[z ArcSin[z]/(1+z)^2, {z, 0, 1}]
> -Infinity
>

This is performance-dependent. On my machine:

In[1]:= Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}]

Out[1]= -Infinity

In[2]:= Dynamic[Pause[.5], UpdateInterval -> 1]

In[3]:= ClearSystemCache[]
  ii = Integrate[z*(ArcSin[z]/(1 + z)^2), {z, 0, 1}]

Out[4]= 1/2 (MeijerG[{{-(1/2), 0}, {1}}, {{0, 0, 1/2}, {}}, 1] -
MeijerG[{{-(1/2), 0}, {1, 1}}, {{0, 0, 1/2, 1/2}, {}}, 1]/Sqrt[Pi])

The MeijerG expression is correct, the issue is that the simplifier
goes astray:

In[5]:= Cases[ii, _MeijerG, -1] // FunctionExpand

Out[5]= {-Infinity, MeijerG[{{-(1/2), 0}, {1, 1}}, {{0, 0, 1/2, 1/2},
{}}, 1]}

As a workaround, you can evaluate the MeijerG[]s as limits:

In[6]:= ii /. HoldPattern@ MeijerG[s__, 1] :>
  Limit[FunctionExpand@ MeijerG[s, z], z -> 1] // Simplify

Out[6]= -1 - 2 Catalan + Pi (1/4 + Log[2])

Maxim Rytin
m.r at inbox.ru



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