I read about a nice way to think about that troubling complex logarithm function. Maybe some of you know this but I think it is helpful.

Given a>0 we know that \ln a = b if and only if e^b = a. Now given a complex number z\not =0 we want to think about \ln z=b. Meaning e^b = z. But note that e^{b+2\pi n i} = e^b \cdot e^{2\pi n i}=e^b. So b+2\pi n i also work. Which leads to a problem. It is not a function!

So one way to fix this is to restrict ourselves (as with inverse sine and cosine functions). First \mbox{arg}(z) is the "argument" of z it is defined to be the angle z creates in the complex plane with 0\leq \theta <2\pi. (Some people use the interval -\pi < \theta\leq \pi). To avoid confusion I will use \ln as the standard function defined for real variables and \log as its complex analogue. We given a complex number z\not = 0 we define,
\log z = \ln |z| + i\cdot \mbox{arg}(z). Note that is z is real and positive it produces the same thing as \ln.

The terrible thing is that the nice properties about \ln do not generalize to \log. For example, \log (z_1z_2) \not = \log (z_1) + \log (z_2).

This is were the "multifunction" comes in. We define a multifunction as \mathbb{C}\mapsto \mathcal{P}(\mathbb{C}). That is, a function which sents a complex number into a subset of the complex numbers. Hence it is a function, not a complex function, but it is a function. So we can define \mbox{Log}(z) = \{ \log z + 2\pi n i\} (note the captial L). Meaning all the solutions to the equation e^x = z.
The good think about this is that the logarithm property is preserved. That is \mbox{Log}(z_1z_2) = \mbox{Log}(z_1)+\mbox{Log}(z_2). (Addition here means adding terms in the set together).