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Re: troubling simple integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91371] Re: troubling simple integral
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 19 Aug 2008 07:13:21 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g86844$hr7$1@smc.vnet.net> <g88v87$gva$1@smc.vnet.net> <g8b8tv$8qa$1@smc.vnet.net>

did wrote:
> On Aug 17, 12:41 pm, Jean-Marc Gulliet <jeanmarc.gull... at gmail.com>
> wrote:
>> Does the following result look better?
>>
>> In[1]:= Assuming[x > 0 && Im[a] == 0 && Im[b] == 0 && Im[y] ==
> = 0,
>>   FullSimplify[
>>    Integrate[(b + k*x)/k^2*Exp[-k*x]*Sin[k*a]*Sin[k*y], {k, 0,
>>      Infinity}]]]
>>
>> Out[1]= 1/4 (2 b ((-a + y) ArcTan[(a - y)/x] + (a + y) ArcTan[(a + y)/
>>          x] - x ArcTanh[(2 a y)/(a^2 + x^2 + y^2)]) +
>>     x Log[1 + (4 a y)/(x^2 + (a - y)^2)])
> 
> That looks much better indeed. Thanks.
> 
>> Note that *Assuming* passes the assumptions to both Integrate[] *and*
>> FullSimplify[]. (In your original expression, only Integrate[] could
>> take into account the assumptions.)
> 
> I understand that. But what I find puzzling the result obtained by Alberto
> (I got the same). I'm wondering if there is an issue there.

The issue might be platform specific. On my system 64-bit Intel Core 2 
Duo 4 GB RAM Mac OS X Leopard 1.5.4 Mathematica 6.0.3, Alberto Verga's 
expression returns unevaluated after few dozens of seconds.

   In[1]:=
     Integrate[1/k^2*Exp[-k]*Sin[a k] Sin[y k],
       {k,0,Infinity},Assumptions->Element[{a,y},Reals]]

   Out[1]=
     Integrate[(E^-k Sin[a k] Sin[k y])/k^2,
       {k,0,\[Infinity]},Assumptions->(a|y)\[Element]Reals]

   In[2]:= $Version

   Out[2]= 6.0 for Mac OS X x86 (64-bit) (May 21, 2008)

Regards,
-- Jean-Marc



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