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Re: More /.{I->-1} craziness

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106300] Re: More /.{I->-1} craziness
  • From: David Bailey <dave at removedbailey.co.uk>
  • Date: Thu, 7 Jan 2010 02:34:34 -0500 (EST)
  • References: <200912300915.EAA17299@smc.vnet.net> <hhhmn8$o9t$1@smc.vnet.net> <hhkjc4$5ag$1@smc.vnet.net> <hhpl4j$9mf$1@smc.vnet.net> <421988.1262677154744.JavaMail.root@n11> <005001ca8e15$766e7bd0$634b7370$@net> <hi1qeg$enm$1@smc.vnet.net>

Richard Fateman wrote:
> I think it is interesting that the same issue came up in the design of 
> another computer algebra system, years ago.
> That is, which objects are "atomic" and which are decomposable for 
> purposes of substitution.  And further,
> of those which are decomposable, how much cleverness should be applied 
> during substitution
> 
> For example,  Exp[I x] -Exp[- I x]  /.   Exp[I x] -> s   should probably 
> result in s-1/s.
> In Mathematica, one gets  s-E^(-Ix).
> 
> can either
> (1) Make this come out s-1/s
>   or
> (2) Argue that Mathematica already does the right thing, blame the user, 
> blame the documentation, blame the nature of mathematics, claim that it 
> is impossible to "read the user's mind" etc.
> 
> To me, the question is simply, by what programming technique can we make 
> Mathematica do the truly expected thing.

Notice that using the transformation rule Exp[I x] -> s (or f[x]->s in 
general) in the way you require, involves inverting it to produce 
x->g[s] for some g. In general g may not be unique, which is why the 
following code generates a warning, but essentially does what you want 
to do:

Solve[Eliminate[{ans == Exp[I x] - Exp[-I x], Exp[I x] == s}, {x}], ans]

Reduce (rather than Eliminate) yields a more mathematically precise 
answer, but the result is considerably more clumsy.

Note also that ReplaceAll *can* be used to do mathematical operations 
without complications provided the LHS of each rule is a variable (but 
not a constant such as I, Pi, etc).

David Bailey
http://www.dbaileyconsultancy.co.uk


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