a challenging definite integral!
- To: mathgroup at smc.vnet.net
- Subject: [mg79317] [mg79317] a challenging definite integral!
- From: dimitris <dimmechan at yahoo.com>
- Date: Tue, 24 Jul 2007 06:09:21 -0400 (EDT)
This integral has appeared some time ago
in a Vladimir Bondarenko post in another forum.
Occasionally, it has appeared in some diferent
posts also in MathGroup.
Mathematica 5.2 and 6 gives incorrectly
Pi^2/6 while another CAS returns unevaluated
the integral.
In[260]:=
Integrate[Log[1 + Exp[-z]/Sqrt[z]], {z, 0, Infinity}]
{N[%], NIntegrate[Log[1 + Exp[-z]/Sqrt[z]], {z, 0, Infinity}]}
Out[260]=
Pi^2/6
Out[261]=
{1.6449340668482262, 1.0179913870581465}
convert("Integrate[Log[1 + Exp[-z]/Sqrt[ z]], {z, 0, Infinity}]
",FromMma,evaluate);evalf(%);
infinity
/
| exp(-z)
| ln(1 + -------) dz
| 1/2
/ z
0
1.017991387
In Mathematica the integral is evaluated (mistakenly as we said) by the
convolution
method.
At this point I would like to show Professor's Richard Fateman
invaluable
comments about this integral on a private communication.
"...While it may not seem hard to you, it turns out that there are
three
components to the Risch integration algorithm: exponential, log and
algebraic.
this integrand, if it is done using the Risch algorithm, requires
that
all of them work, together.
Since there is not, so far as I know, a complete implementation of
the
algebraic case in Mathematica there may be problems with
the hacks taken to bypass this lack of implementation.
So I don't find the issue to be surprising..."
I have spent considerable time for this integral!
It is not an important task its evaluation, but I would like to see
possible workarounds (by hand or/and CAS in
order to get a closed form solution.
Thank you very much!
Dimitris S. Anagnostou