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MathGroup Archive 1997

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Re: Re: error in complex integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg7449] Re: [mg7412] Re: [mg7372] error in complex integration
  • From: seanross at worldnet.att.net
  • Date: Wed, 4 Jun 1997 02:30:36 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Timo Felbinger wrote:
> 
> 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


You are always free to write your own routines.  The primary benefit of
mathematica is not all the built in routines, but the underlying
structure of the language which, in my opinion, is far easier and
flexible than either C or Fortran.  As far as excessive claims, it is a
disease that affects advertising agencies.  I have never purchased a
software product that did not make slightly to greatly exaggerated
claims.


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