Imaginary Exponent : (x^1i)^1i

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.

Which (x^i)^i are you asking for - we have an infinite number of them - see my other post.

But for every of those hyperbolas is the answer the same - obviously Min and Max are +/- infinity

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".