Hi,
You mentioned that "the integrand is not Lebesgue integrable on R" which I would call just not in L^2(R). Yes I noticed that but I
didn't really expect an error message like the one I received. It should have said that the integral does not converge or something
like that. Also you said that "%o2 shows that one needs to be careful in using the output of maxima's computations--maxima told you
that its answer is valid on t>0 and made no promises about t<=0.". I would like to say about this that the answer is not clear
about the domain of t because sometimes when Maxima asks a question about the sign of a variable the answer is the same no matter
what you say and so it is true for all t, but I would prefer an answer that is valid for all t even if you have to use piecewise
functions in the answer. The fact that piecewise solutions are (or seem to be) banned from Maxima is a strange weakness in Maxima
(and I suspect other CAS also). abs(-t) = pw([minf,t,0,-t,inf],t) so even abs(t) seems to be or is treated as if it is not a good
answer.
Rich
On Sun, 24 May 2009, Dieter Kaiser wrote:
< Am Samstag, den 23.05.2009, 23:21 -0600 schrieb Robert Dodier:
< > On 5/22/09, Richard Hennessy <rich.hennessy at verizon.net> wrote:
< >
< > > (%i1) integrate(1/(x^2+1)*exp(-%i*%pi*x*t),x,minf,inf);
< > > Is t positive, negative, or zero?
< > > p;
< > > (%o1) %pi*%e^(-%pi*t)
< > > (%i2) integrate(1/(x^2+1)*exp(-2*%i*%pi*x*t),x,minf,inf);
< > > Is t positive, negative, or zero?
< > > p;
< > > (%o2) %pi*%e^(- 2*%pi*t)
A mathematical comment for RH:
Note that %o2 is correct only for t>=0. The correct answer for all t
requires abs(t) in place of t. Indeed, the modulus of %o2 should be bounded by 0 and
%pi for all t and be in L^2(R). %o2 shows that one needs to be careful
in using the output of maxima's computations--maxima told you that its
answer is valid on t>0 and made no promises about t<=0.
Also, %i3 will produce rubbish (the integrand is not Lebesgue
integrable on R).
< > > (%i3) integrate(%*exp(2*%i*%pi*x*t),t,minf,inf);
I am sure that you know this, but since you did not mention it in your
post, I thought I should point it out.
Leo
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