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Re: A FullSimplify Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg41074] Re: A FullSimplify Problem
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 1 May 2003 04:58:38 -0400 (EDT)
  • References: <b8o20k$p28$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Ted,
Sorry, I made a mistake. Please ignore my previous answer.

Recently, in the thread "Simplification of definite integral?" in this 
group I initiated a discussion on a similar problem with Integrate. Here 
even wrong results were produced. The problem seems to be connected to 
the integrand having a branch cut singularity (as e.g. Log, Sqrt have).

You can study this by looking at the simplified version of your 
integral, viz.

In[1]=Integrate[Log[1-2a x +a^2],{x,0,1}]

starting with the general integral

In[2]=Integrate[Log[1-2a x + a^2], x]

Mathematica gives

Out[2] =

                 2
-x + x Log[1 + a  - 2 a x] +

          2            2
   (-1 - a ) Log[-1 - a  + 2 a x]
   ------------------------------
                2 a

In the last Log we observe a negative sign of the term, i.e. mathematica 
has (unmotivatedly??) moved to another branch of the Log-function.

Forming the derivative of Out[2] gives back the integrand, but the 
derivative of the function with the positive sign under the second Log 
gives the same result.

Hope this helps a bit. Summa summarus: mathematica seems to have a lot 
of difficulties here.

Regards,
Wolfgang

Ersek, Ted R wrote:

> At  http://mathworld.wolfram.com/LeibnizIntegralRule.html  
> I learned that 
>    Integrate[Log[1-2a Cos[x]+a^2],{x,0,Pi}]
>    = 2*Pi*Log[Abs[a]]
> 
> Mathematica knows how to do this integral, but gives a much more complicated
> result.  Can anyone explain how to use FullSimplify and other
> transformations to show that the complicated result Mathematica gives is
> equivalent to the answer above?
> 
> Thanks,
>    Ted Ersek
> 
> 
> 



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