Re: limit challenge, simple
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 18 Aug 2006 19:08:55 GMT
In article <1155871025.440141.201870@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
rjf <fateman@xxxxxxxxx> wrote:
Again, it is not difficult to come up with examples that show flaws in
elementary components of CAS. No need to conflate their failures to
inadequate programs for definite integration, when the problems are
much more fundamental.
Consider the value of exp(2*pi*i*x) as x-> infinity. (real). Most
people with some familiarity with a course in complex variables will
realize that this expression just spins around the origin.
So, of course, does exp(i*x), just a little bit slower.
And exp(i*log(x)) does the same thing, even slower. But it still spins
around the origin.
That last expression can be written as x^i.
Examine the limit(x^i, x-->infinity) in various systems. The limit is
NOT the (real) interval [-1,1], as claimed by Mathematica 5.1. Maple
refuses this one. Macsyma says "undefined".
Maple 10 returns -1-I .. 1+I for limit(exp(I*ln(x)), x=infinity)
and for limit(evalc(x^I), x=infinity).
It is also possible to consider abs(x^i). Its limit should be 1. (OK
in Mathematica 5.1;
in Maple 7, the newest version I have at my disposal, it comes out
[0..sqrt(2)]!) Macsyma says 1.
In Maple, limit(abs(exp(I*ln(x))), x=infinity) returns 1, as does
simplify(evalc(abs(x^I))) assuming x > 0.
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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