For complex numbers define,

$\exp (z)=\sum _{n=0}^{\infty }{z}^n/n!$

For $w$ not equal to 0 define

$w^z=\exp (z\log (w))$

$=\exp [z(\ln |w|+i(\arg w+2n\pi ))]$

(any integer $n$).

So if $w=e$ then $e^z=\exp (z)\exp (2n\pi \mathrm{iz})$ has many values. Take $n=0$ for the convention.

$z^w=e^{w(\log z)}$

where we are taking logarithms to the base $e$. There are several things we have to consider here:

1) What it means to take the natural logarithm of a complex number - if we write $z=r{e}^{i\theta }$ (where $r$ is real and $\theta$ is the argument of $z$), then we may write that

$\log z=\log (r{e}^{i\theta })=\log r+\log e^{i\theta }=\log r+i\theta$,

2) Note how we have to be careful with this, because the above complex logarithm is multivalued (i.e. since $e^{2i\pi }=1$, then we could add any multiple of $2\pi$ to $\log z$). Hence we restrict the argument of $z$ to be within some range of width $2\pi$ (i.e. encompassing all of the complex plane), but such that there is no overlap, i.e. we may let (for example)

$-\pi <\theta \le \pi$

Such a restriction of the argument of $z$ is known as a branch cut (in fact, the above branch is known as the principal branch). Another possibility would be $0\le \theta <2\pi$.

The whole point of this is to prevent the function $z\to z^w$ being multivalued (i.e. the function is well defined). However, the range of $\theta$ which we choose is purely arbitrary; in fact, different branches define different functions.

However, it is relatively easy to show that this definition coincides with the (obviously single value) $z^n$ for integer $n$ (in this case, the branch cut, since a change in the argument by $2\pi$ does not change $z^n$, as $e^{2i\pi }=1$).