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 we know that if and only if . Now given a complex number we want to think about . Meaning . But note that . So 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 is the "argument" of it is defined to be the angle creates in the complex plane with . (Some people use the interval ). To avoid confusion I will use as the standard function defined for real variables and as its complex analogue. We given a complex number we define,
. Note that is is real and positive it produces the same thing as .
The terrible thing is that the nice properties about do not generalize to . For example, .
This is were the "multifunction" comes in. We define a multifunction as . 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 (note the captial L). Meaning all the solutions to the equation .
The good think about this is that the logarithm property is preserved. That is . (Addition here means adding terms in the set together).