How Many Roots Exist for Complex Numbers Raised to Irrational Powers?

[cjellison]

There are n nth roots to every complex number (except zero).

My question: How many "roots" are there when you take a complex number to an irrational or transcendental number. For that matter, how do we define raising a number to an irrational number? How do we define raising a number to a transcendental number?

Hmm...how is this defined? By a taylor series?

$$e^{1/e}$$

 

[Hurkyl]

By definition, in the complexes:

$$z^w := \exp(w \mathop{\mathrm{Log}} z)$$

 

[HallsofIvy]

In the complex plane, a number to an irrational (whether transcendental or not) power has an infinite number of values.