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MathGroup Archive 2005

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Re: Version differences in Integrate with Min[] in the integrand

  • To: mathgroup at smc.vnet.net
  • Subject: [mg55913] Re: Version differences in Integrate with Min[] in the integrand
  • From: Maxim <ab_def at prontomail.com>
  • Date: Sat, 9 Apr 2005 03:56:59 -0400 (EDT)
  • References: <d355tt$ope$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Fri, 8 Apr 2005 05:47:41 +0000 (UTC), Gareth Russell  
<gjr2008 at columbia.edu> wrote:

> Hi,
>
> I don't think this is a function of differences in the internal use of
> Simplify. Anyone have an idea about what is going on here?
>
> Gareth Russell
> Columbia University
>
>
> Integrate[Exp[-r*(Min[d2,t1]-Min[
>         d1,t1])]/((d12-d11)*(d22-d21)),{d1,d11,d12},{d2,d21,d22}]
>
> The result in v4.2
>
> \!\(\(\_d11\d12\(\_d21\d22\( \[ExponentialE]\^\(\(-r\)\ \((\(-Min[d1,  
> t1]\
> \) + Min[d2, t1])\)\)\) \[DifferentialD]d2 \
> \[DifferentialD]d1\)\)\/\(\((\(-d11\) + d12)\)\ \((\(-d21\) + d22)\)\)\)
>
> The result in v5.0
>
> \!\(\(\[ExponentialE]\^\(\(-d21\)\ r\)\ \((\(-\[ExponentialE]\^\(d11\ r\)\)  
> - \
> \[ExponentialE]\^\(d21\ r\) + \[ExponentialE]\^\(r\ \((d11 + d21 -  
> t1)\)\) + \
> \[ExponentialE]\^\(r\ t1\))\)\)\/\(\((d11 - d12)\)\ \((d21 -  
> d22)\)\ r\^2\)\)
>
> (Incorrect, I think...)
>
> The result in v5.1.0
>
> Nothing: runs for many minutes.
>

In such cases it is a good idea to integrate piecewise terms or factors  
separately:

Assuming[d11 < d12 && d21 < d22,
   Integrate[Exp[r*Min[d1, t1]], {d1, d11, d12}]*
     Integrate[Exp[-r*Min[d2, t1]], {d2, d21, d22}]/
       ((d12 - d11)*(d22 - d21))]

This evaluates almost instantly in version 5.1. For some reason Integrate  
takes a very long time (keeps running after 10 minutes) without an  
assumption implying that the limits of integration are real.

Maxim Rytin
m.r at inbox.ru


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