Re: Why does this happen?
- To: mathgroup at smc.vnet.net
- Subject: [mg78786] Re: Why does this happen?
- From: dimitris <dimmechan at yahoo.com>
- Date: Mon, 9 Jul 2007 01:43:35 -0400 (EDT)
- References: <f6ks38$lbm$1@smc.vnet.net><f6qeom$aba$1@smc.vnet.net>
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