
Multifunctions
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 $\displaystyle a>0$ we know that $\displaystyle \ln a = b$ if and only if $\displaystyle e^b = a$. Now given a complex number $\displaystyle z\not =0$ we want to think about $\displaystyle \ln z=b$. Meaning $\displaystyle e^b = z$. But note that $\displaystyle e^{b+2\pi n i} = e^b \cdot e^{2\pi n i}=e^b$. So $\displaystyle b+2\pi n i$ 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 $\displaystyle \mbox{arg}(z)$ is the "argument" of $\displaystyle z$ it is defined to be the angle $\displaystyle z$ creates in the complex plane with $\displaystyle 0\leq \theta <2\pi$. (Some people use the interval $\displaystyle \pi < \theta\leq \pi$). To avoid confusion I will use $\displaystyle \ln $ as the standard function defined for real variables and $\displaystyle \log $ as its complex analogue. We given a complex number $\displaystyle z\not = 0$ we define,
$\displaystyle \log z = \ln z + i\cdot \mbox{arg}(z)$. Note that is $\displaystyle z$ is real and positive it produces the same thing as $\displaystyle \ln$.
The terrible thing is that the nice properties about $\displaystyle \ln $ do not generalize to $\displaystyle \log$. For example, $\displaystyle \log (z_1z_2) \not = \log (z_1) + \log (z_2)$.
This is were the "multifunction" comes in. We define a multifunction as $\displaystyle \mathbb{C}\mapsto \mathcal{P}(\mathbb{C})$. 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 $\displaystyle \mbox{Log}(z) = \{ \log z + 2\pi n i\}$ (note the captial L). Meaning all the solutions to the equation $\displaystyle e^x = z$.
The good think about this is that the logarithm property is preserved. That is $\displaystyle \mbox{Log}(z_1z_2) = \mbox{Log}(z_1)+\mbox{Log}(z_2)$. (Addition here means adding terms in the set together).