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MathGroup Archive 1997

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Re: error in complex integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg7412] Re: [mg7372] error in complex integration
  • From: Timo Felbinger <felbing at pandora.physik.uni-konstanz.de>
  • Date: Sat, 31 May 1997 15:07:21 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com


On Thu, 29 May 1997, Richard Finley wrote:

> Timo,
> 
> Integrating in the complex plane is always fraught with difficulties because
> one must pay close attention to the domains of the
> functions....unfortunately, one cannot yet expect a package like Mma to
> automatically keep track of this and that is the problem you have here.  The
> reason for your difficulty will become clear if you integrate your complex
> expression symbolically...you will see that the answer involves Log and Log
> has a branch cut from (0, -Infinity).  If you carefully evaluate the result
> of the symbolic integration and take account of the different values of Log
> as it approaches the branch cut from above and below ( which gives a factor
> of 2 Pi), you will get the correct answer. 
> 
Well, yes. But a situation like this may easily occur inside a complicated
expression, without me even knowing about it, rendering all answers given 
by Mma virtually useless!

As far as I know, being unable to handle branch cuts does not allow you 
to neglect them, at least from a mathematicians point of view (physicists 
may sometimes hold a different attitude ;-)  ). To put it straight: In my 
opinion, a package with a name derived from the word 'mathematics' should 
never make a statement which it cannot prove to be true, at least unless one
explicitely requests the use of numerical methods (or, maybe, it could
provide a global switch to choose between 'hazardous' and 'cautious'
mode). 

Why does Mma not just return an expression with all the Log's still in it,
admitting that it cannot handle functions which may involve branch
cuts in a safe way, and let me figure out the correct result on my
own? 


Timo Felbinger



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