Re: Why does this happen?
- To: mathgroup at smc.vnet.net
- Subject: [mg78798] Re: Why does this happen?
- From: dreiss at scientificarts.com
- Date: Tue, 10 Jul 2007 06:23:30 -0400 (EDT)
- References: <f6ks38$lbm$1@smc.vnet.net><f6sim4$8bn$1@smc.vnet.net>
yes indeed... hence my post.... On Jul 9, 1:56 am, dimitris <dimmec... at yahoo.com> wrote: > David Reiss : > > > > > OK, since most folks didn't catch Budasoy's typo in the Exp. Here is > > an "analysis" of the problem (Mathematica 6.01. There does appear to > > be a numerical inconsisstency between the exact result and the > > numerical one. Is this possibly due to the singularity of the > > integrand at 0? Or perhaps we have a bug... 'tis not clear to me > > before my morning coffee... > > > (M 6) In[1]:= Integrate[Log[1 + Exp[-x]/Sqrt[x]], {x, 0, Infinity}] > > > (M 6) Out[1]= \[Pi]^2/6 > > > (M 6) In[2]:= Limit[Log[1 + Exp[-x]/Sqrt[x]], x -> Infinity] > > > (M 6) Out[2]= 0 > > > (M 6) In[3]:= Limit[Log[1 + Exp[-x]/Sqrt[x]], x -> 0] > > > (M 6) Out[3]= \[Infinity] > > > (M 6) In[4]:= N[\[Pi]^2/6] > > > (M 6) Out[4]= 1.64493 > > > (M 6) In[5]:= Table[ > > NIntegrate[Log[1 + Exp[-x]/Sqrt[x]], {x, 10^-n, n 10}], {n, 1, 10}] > > > (M 6) Out[5]= {0.837883, 0.989369, 1.01402, 1.01748, 1.01793, \ > > 1.01798, 1.01799, 1.01799, 1.01799, 1.01799} > > > (M 6) In[6]:= Integrate[Log[1 + Exp[-a x]/x^(1/n)], {x, 0, Infinity}, > > Assumptions -> {Re[1/n] < 1, a > 0}] > > > (M 6) Out[6]= (n \[Pi]^2)/(12 a (-1 + n)) > > > (M 6) In[7]:= (n \[Pi]^2)/(12 a (-1 + n)) /. {n -> 2, a -> 1} > > > (M 6) Out[7]= \[Pi]^2/6 > > > (M 6) In[8]:= quickanddirty[delta_] := > > Module[{data}, > > > data = Table[ > > N@Log[1 + Exp[-x]/Sqrt[x]], {x, 10^-5, 10, delta}]; > > > Tr[data delta] > > ]; > > > (M 6) In[9]:= quickanddirty[10^-2] > > > (M 6) Out[9]= 1.05924 > > > (M 6) In[10]:= quickanddirty[10^-3] > > > (M 6) Out[10]= 1.02151 > > > (M 6) In[11]:= quickanddirty[10^-4] > > > (M 6) Out[11]= 1.01823 > > > On Jul 6, 3:47 am, Budaoy <yaomengli... at gmail.com> wrote: > > > I have a problem in calculating this integral shown below: > > > > Integrate[Log[1+Exp[x]/Sqrt[x]],{x,0,Infinity}] > > > Pi^2/6 > > > > N[%] > > > 1.64493 > > > > NIntegrate[Log[1+Exp[x]/Sqrt[x]],{x,0,Infinity}] > > > 1.01799 > > > > Where does this difference come from and which one is correct? > > It is a bug in Integrate. > NIntegrate's result is correct! > > Dimitris