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Integral. Wrong answer.

Iurii Popov
Posted 12 years ago
POSTED BY: Iurii Popov
6 Replies
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POSTED BY: Frank Kampas

Yeah, looks like a bug (analyzing for convergence is a difficult area to automate). Will investigate further.
POSTED BY: Daniel Lichtblau
Iurii Popov
Iurii Popov
Posted 12 years ago

As I can see the bug has not been fixed yet. Do you know the class of functions that we can believe the result of integration? I am afraid to use Mathematica because of uncertain results.
POSTED BY: Iurii Popov

It was fixed in the internal version under development. It was not back-ported to 10.0.1. It is not considered to be a show-stopper level of bug and, to my mind, the risks of doing so outweigh the benefits. Risks include both delays in shipping, and the possibility of introducing subtle new bugs.
POSTED BY: Daniel Lichtblau

In[10]:= Integrate[(1 - Exp[-x])/x Cos[x], {x, 0, ul}, Assumptions -> {ul > 0}]

Out[10]= 1/2 (2 CosIntegral[ul] - ExpIntegralEi[(-1 - I) ul] - ExpIntegralEi[(-1 + I) ul] + Log[2])

In[11]:= Limit[%, ul -> [Infinity]]

Out[11]= 1/4 (Log[-2 - 2 I] - Log[-1 - I] - Log[-1 + I] + Log[-(1/2) + I/2] - Log[1/2 - I/2] - Log[1/2 + I/2] + Log[1 - I] + Log[1 + I])

In[12]:= FullSimplify[%]

Out[12]= Log[2]/2
POSTED BY: Frank Kampas

You can obtain the result using the PrincipalValue -> True option. This setting tell Integrate to handle simple poles.

$Version
Integrate[(1 - Exp[-x])/x Cos[x], {x, 0, Infinity}, PrincipalValue -> True]

enter image description here

Btw, Version 9 also handles it the same way (gives same error without the PrincipalValue -> True setting) so this might not be a bug, but I am not an expert on this.
POSTED BY: Nasser M. Abbasi
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