Is it possible to define irrational powers for negative numbers?

[Unredeemed]

I've been trying to plot the graph of y=x^(1/x).

The positive values of x have been fine, but the negative values have presented quite a challenge.

For even negative integers, I realized the y value was complex and for odd negative integers it was real.

Then, I started thinking about the vaues between integers (e.g. -2.5^(1/-2.5) and could only really come up with an answer for the rational negatives.

Am I right in thinking that you can find out by transforming the number into an integer by multiplying by a power of 10 and then seeing if it's odd or even?

I had absolutely no idea about how to deal with the irrational number (e.g. -pi^(-pi)). Does it require some kind of definition for irrational powers?

Thanks, in advance, for your help.

Unredeemed.

 

[mathman]

One approach is to use the following: x^y=e^yln(x).

If x < 0 (or in general, x complex), x=rexp(iu). So ln(x)= ln(r) +iu. For x < 0, u=π.

Therefore for x < 0, x^y=e^{y{ln(|x|)+πi}}.

I hope this helps!

 

[HallsofIvy]

Generally speaking, the function $f(x)= a^x$ is only defined for positive a. That means that $f(x)= x^x$ can only be defined for positive x.

(Or, as Mathman does, go into complex numbers.)