monir wrote...
Thank you all for your thoughtful replies. Here're some comments:

1) One would expect the w/s function IMPOWER("x+yi",1/3) to return the
principal value, if any, similar to:
....=IMSQRT("-4+0i") correctly returning +2i and not -2i
....=IMPOWER("8+0i",1/3) correctly returning +2.0, and not -1+sqrt(3)i or
-1-sqrt(3)i

From the World of Mathematics: 'Informally, the term "principal root"
is often used to refer to the root of unity having smallest positive
complex argument.' In this sense, Excel's IMPOWER *does* return the
principal root.

2) Hence, IMPOWER("-8+0i",1/3) should return -2.0 and not one of the
conjugate pair 1+sqrt(3)i or 1-sqrt(3)i.

'Hence' based on a faulty (in this case, faulty semantics) conditions
leads to vacuous conclusions. Notationally, A = B, if A is false the
statement is true no matter whether B is true or false.

3) My experience with complex numbers in XL environment is rather limited.
However, one might reasonably argue that a complex number with zero imaginary
coefficient is equivalent to a real number!

Yup. So?

Now try =POWER(-8,1/3). You would correctly get -2.0 and not #NUM!

Again, so? *All* reals have real odd order roots. If you want real
roots for real numbers, use POWER, not IMPOWER.

4) IMPOWER() actually relies on ATAN2(x,y) and not ATAN(y/x) to convert
complex numbers to polar, contrary to the XL Help info on the function.
....

So? ATAN2 and equivalents in other programming languages are
**ALWAYS** preferable to ATAN. Anyway, online help for IMPOWER doesn't
show ATAN, it shows a symbolic inverse tangent. Not unambiguous, but
not necessarily an error.

This together with the always non-negative "r" maybe an internally-wired
factor determining which value is returned by IMPOWER() and similar complex
number functions.

If you're talking about polar coordinates, r is necessarily always
nonnegative BY DEFINITION. Norms always give nonnegative real results.
And the principle nth root is always based on the principal argument
in the interval [0, 2 pi) and dividing it by n. That ALWAYS returns
the root in the 1st quadrant of the complex plane with the smallest
positive argument, so the principal root as informally defined above.

In this case Microsoft is following generally accepted mathematical
conventions. Adapt!