#296 limit(exp(x*%i)*x,x,inf) should give infinity

[Stavros Macrakis]

limit(exp(x*%i)*x,x,inf) => UND NO!
Should be INFINITY

[Barton Willis]

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Let F : R -> C and F(x) = x exp(i x) = x cos(x) + i x sin(x).
Both the real and imaginary parts of F are oscillatory with
linearly growing amplitudes; neither the real nor the imaginary
parts have a limit towards infinity. I say the limit is UND.

Barton

[Stavros Macrakis]

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I believe that the definition of limit(f(x))=infinity is that for all N,
there exists an X such that x>X implies abs(f(x))>N.

That is satisfied in this case. In fact, you can choose X=N.
The separate magnitudes of the real and imaginary parts are
irrelevant.

After all, limit(2+x*%i,x,inf) = infinity

[Dieter Kaiser]

The example of this bug report gives no longer 'und but a noun form (Maxima 5.20post);

(%i11) limit(exp(x*%i)*x,x,inf);
(%o11) 'limit(x*%e^(%i*x),x,inf)

I am not sure what is the right answer. Wolfram alpha gives a result in terms of an interval:

E^((2 I) Interval[{0, Pi}]) Infinity

A noun form is not a wrong result. Perhaps, we can close this bug report at this point. Further improvements of the limit routines might give a more complete answer.
Setting the status to pending and the resolution to "works for me".

Dieter Kaiser

[Stavros Macrakis]

A noun form is certainly better than und, but the correct result is Infinity. I would have thought that in the Wolfram world, the correct result would be ComplexInfinity (which corresponds to Maxima's Infinity).

[Dieter Kaiser]

Changing the title to reflect the issue better. Setting the resolution back to "None".
Dieter Kaiser

[Barton Willis]

I'm fairly sure this is a bug in simplimtimes:

(%i7) limit(exp(x*%i)*x,x,inf);

  0: (MAXIMA::SIMPLIMTIMES
      (((MAXIMA::MEXPT MAXIMA::SIMP MAXIMA::IRREDUCIBLE) MAXIMA::$%E
        ((MAXIMA::MTIMES MAXIMA::SIMP MAXIMA::RATSIMP) MAXIMA::$%I MAXIMA::$X))
       MAXIMA::$X))
  0: SIMPLIMTIMES returned $UND
(%o7)                                 und

The function simplimtimes apparently believes that inf x ind = und is an identity.

I also think that is bug is responsible for at least one other testsuite bug.