Bug report ID: 719832 - limit(exp(x*%i)*x, x, inf) should give infinity

ind*inf is only infinity if the ind doesn't include 0, as shown by the
sin(x)*x example, which should be und, not infinity.

On 2010-04-06, Dieter Kaiser <drdieterkaiser at web.de> wrote:
> We have the open bug report ID: 719832 - limit(exp(x*%i)*x,x,inf) should
> give infinity.
>
> Maxima knows the following limits:
>
> (%i5) limit(exp(%i*x),x,inf);
> (%o5) ind
> (%i6) limit(x*ind,x,inf);
> (%o6) infinity
>
> But Maxima does not evaluate the reported example to infinity:
>
> (%i7) limit(x*exp(%i*x),x,inf);
> (%o7) und
>
> In the routine simplimtimes we have code to check for an '$ind in a
> product. For this case Maxima always returns the result '$und. We might
> add code to check for the case '$ind*'$inf -> '$infinity:
>
> ...
>   ((equal num 1)
>    (setq sign ($csign prod))
>    (return (cond ((and flag2
>                        (member flag '($inf $minf)))
>                   ;; Check in addition the case ind*inf -> infinity.
>                   '$infinity)
>                  (flag2 '$und)
> ...
>
> With this extension we will get the desired result for the example of
> the bug report:
>
> (%i1) limit(exp(%i*x)*x,x,inf);
> (%o1) infinity
>
> We get the same result for the following:
>
> (%i2) limit(exp(-%i*x)*x,x,inf);
> (%o2) infinity
>
> And as a consequence:
>
> (%i3) limit(sin(x)*x,x,inf);
> (%o3) infinity
>
> (%i4) limit(cos(x)*x,x,inf);
> (%o4) infinity
>
> Two integrals in rtestint.mac will get a problem:
>
> ********************** Problem 59 ***************
> Input:
> integrate(cos(9*x^(7/3)),x,0,inf)
>
>
> Result:
> Principal Value
> 0
>
> This differed from the expected result:
> 3*gamma(3/7)*cos(3*%pi/14)/(7*9^(3/7))
>
> ********************** Problem 60 ***************
> Input:
> integrate(sin(9*x^(7/3)),x,0,inf)
>
>
> Result:
> 'integrate(sin(9*x^(7/3)),x,0,inf)
>
>
> The reason is, that Maxima no longer will get the limit '$und but
> '$infinity in the routine limit-pole for the involved trig expressions,
> e.g.
>
> (%i2) limit(x*cos(9*x^2),x,inf);
> (%o2) infinity
>
> But the routine limit-pole does not handle the case '$infinity and
> returns the default answer '$yes and not the expected result '$und. This
> could be corrected in the routine limit-pole by adding the case
> '$infinity with a result '$no:
>
> (defun limit-pole (exp var limit direction)
>   (let ((ans (cond ((member limit '($minf $inf) :test #'eq)
>                     (cond ((eq (special-convergent-formp exp limit)
> '$yes)
>                            '$no)
>                           (t (get-limit (m* exp var) var limit
> direction))))
>                    (t '$no))))
>     (cond ((eq ans '$no)   '$no)
>           ((null ans)   nil)
>           ((eq ans '$und) '$no)
> --->      ((eq ans '$infinity) '$no)
>           ((equal ans 0.)   '$no)
>           (t '$yes))))
>
> I am not sure about the following two points:
>
> 1. Is it correct to return the result '$infinity for the limits of
>    products with the trig functions sin and cos, e.g. for
>    limit(x*cos(x),x,inf) -> infinity?
>
> 2. The routine limit-pole seems not to expect a result '$infinity.
>    Is it correct in general to return the answer '$no for this case?
>
> Dieter Kaiser
>
>
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