Hmmm, this not only assumes that x is real but its also only one out of an infinite number of "solutions" for (x^i)^i.
In the complex domain we run into ambiguities quite often and easy and rules for calculating with powers etc. not always apply the same way as in domain real. That was the fine thing with MC11 and Maple - it defaults to domain real and most of the time simplified expressions the same way we would do (using some implicit assumptions).
The current Mathcad's muPad can't even simplify (2^i)^i to 1/2 wheras the numerics has no problem
muPad obviously thinks more complex (pun intended):
and so we have multiple ways to express (x^i)^i and Mathcad can't simplify to the one with k=0. I'd wish that MC would at least give the simplification similar to my expression. Mathcad will usually do so with "solve,fully" but in this case with no success
We can "verify" the multiple "solutions" with the numerics (the symbolics again is of no help, possibly again as of other ambiguities)
Just noticed that also Mathcads numerics does NOT simplify that expression to 1/x. Thats the reason for the strange lookin plots with all its jumps
What motivates these questions of yours?
Yes, this would also be of interest to me. Loi Le is around here for some years with his special questions and I also asked few times about the reason or the source of his various problems but unfortunately never got an answer.
I think you dont understand what I meant with ambiguity.
(x^i)^i is not 1 curve but this expression represents an infinite number of curves. The numeric choses one of them depending on th x-value. You have a jump from (1/x)*exp((k-1)*2*pi) to (1/x)*exp((k)*2*pi) at x=exp(2*k-1) for every integer k. So the highest value which Mathcad would yield is exp(pi)=23.141, but thats insignificant. If x approaches zero the "jump points" accumulate.
Don't you think it would be time to tell us where your problems stem from and what you are trying to do?
I see, but isn't there anything you want to achieve? Or is it simply to explore Mathcad and show its sometimes weak symbolics? I guess you knew, or better you thought to known, that (x^1i)^1i ressembles 1/x. As I have shown this isn't true because some calculation rules aren't valid in the domain complex.
Anyway, here is for your information a plot of "your" function (better: relation) with logarithmic scaled abscissa. You see the jumps evenly spread in constant relative distance (in my last post I was missing a pi when I gave the x-values for those jumps; see the file for the correct expression). The blue dotted lines are the various hyperbolas which are all the real graphs of your function. The red curve represents the values Mathcad decided to chose. The study of that curve doesn't make much sense in my opinion.
It seems you didn't really understood my remarks about the mathematical ambiguity of the function in question and the way Mathcad deals with it and I honestly can't think of anything better to explain than the plot in my last post.
I already showed that from the mathematical point of view we have
So your equation has an infinite number of solutions:
Mathcad (practical math??) will give you just the two solutions +1 and -1, depending on the guess value you provide. For some guesses you may even get 23.141 (=exp(1*pi)) because Mathcad ran into some kind of numerical inaccuracy.
I guess that this will be my last reply to this and "related" "problems".
