Re: Integration anomaly?
- To: mathgroup at smc.vnet.net
- Subject: [mg127021] Re: Integration anomaly?
- From: Bill Rowe <readnews at sbcglobal.net>
- Date: Sun, 24 Jun 2012 04:26:50 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
On 6/23/12 at 4:16 AM, john_szumiloski at merck.com (Szumiloski, John)
wrote:
>I recently was playing around with the function Log[1 + 1 / ( t^n )
>], and exploring positive values of n. (I have no interest in
>nonpositive or complex n) In particular, I wanted to look at its
>integral, so I did this: (v8.0.4, Windows XP)
>Integrate[ Log[1+1/(t^n)], { t, 0, Infinity } ]
>which gave:
>
>ConditionalExpression[ -(Pi Csc[Pi/n]), Re[n]<0 ]
I get the same result using
In[7]:= $Version
Out[7]= 8.0 for Mac OS X x86 (64-bit) (October 5, 2011)
>Now I am no analysis expert, but it seems pretty clear to me that
>the integral diverges for negative (real) n.
Not proof but given
In[8]:= Table[NIntegrate[Log[1 + 1/(t^n)], {t, 0, Infinity}],
{n, 5}]
Out[8]= {23233.1,3.14159,3.6276,4.44288,5.3448}
and
In[9]:= Table[NIntegrate[Log[1 + 1/(t^-n)], {t, 0, Infinity}],
{n, 5}]
Out[9]= {5.522502947974294*10^27954,1.104500589594859*10^27955,1.656750884392288*10^27955,2.209001179189718*10^27955,2.761251473987147*10^27955}
it seems clear the integral converges for n<0 and diverges for n>0