Definition of complex exponentiation How come we define complex exponentiation as follows
$z^w = e^{wLogz}$ where $Log(z) = log(|z|) + iarg(z)$. 
I just want the motivation of why they defined it this way.
 A: The use of the exponential function (which is best understood as defined to be the Taylor expansion, so that it is its own derivative) has the same reason as for real exponentiation. This reason means that we need to invert the exponential function . If we restrict to real inputs the inverse exists, but if we allow complex inputs then there is no longer an inverse function due to the periodic nature of the complex exponential function. Thus the definition of the complex logarithm is because you want to choose a particular representative of the many possible inputs on which the exponential function gives a certain output. This is called a branch cut, and your particular definition is just one of many possible branch cuts. We want nice properties such as continuity, but there is no branch cut that is continuous on the whole complex plane, and the next best is to be continuous on the complex plane minus a ray from the origin. If it is the non-positive real axis, as you have in your definition, we call it the principal branch cut.
A: Well, how else would you define it? We know from the real setting that its derivative (if it's differentiable) will have to be 
$$
(z^w)' = \log(z)\,z^w
$$
for some reasonable definition of $\log$. Note that $\log$ is just the obvious thing that works. Clearly $e^{\log(z)}=z$. So we know that we are looking for the solution to the differential equation 
$$
f'(w) = \log(z)f(w)
$$
And if you've taken any differential equations, you know that $e^{\log(z) w}$ is the solution. So that's what we're stuck with.
