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