Logorithms of negative numbers

[SW VandeCarr]

logarithms of negative numbers

Some years ago, I read about how to take the natural log of a negative real number but I don't remember the source. I'd to get feedback if this is correct: (::= 'therefore')

Euler's identity: e^(i pi) = -1 :: ln(-1)= i pi, and ln(-a) = ln(a) + i pi ('a' a real number) :: the natural logs of all negative real numbers would lie on a line parallel to the real axis at distance of pi on the i axis. Is this correct?

 

[rochfor1]

This is possible when you're dealing with complex numbers, but you have to be careful. To be analytic, the complex logarithm requires a branch cut. Traditionally, the principle value of the logarithm places this branch cut along the negative real axis, so using the principle branch of the complex logarithm, it's not possible to take the log of a negative number. However, this branch cut can be moved, resulting in other branches of the complex logarithm. Any half-line terminating at the origin can be used as a branch cut for Log, so if you wanted to take the logarithm of a negative number, you could choose the branch that is cut along the negative imaginary access.

To see this discussion of branch cuts, note that for $$z \in \mathbb{C}$$, the complex logarithm is defined as $$\operatorname{Log} z = \log | z | + i \operatorname{Arg} z$$. The branch cut required to make the complex logarithm analytic comes from the fact that the argument has to jump from 0 to $$2 \pi$$ somewhere, and for the principle branch, that's on the negative real axis. To get a branch of the logarithm that you can use on negative real numbers, simple choose $$-\frac{\pi}{2}\leq\operatorname{Arg} z\leq\frac{3 \pi}{2}$$, which is well-defined on the negative real axis. And in that case your absolutely right, for x < 0, $$\operatorname{Arg} x = \pi$$, so $$\operatorname{Log} x = \log |x| + i \pi$$.

 

[SW VandeCarr]

Thanks very much rochfor1. I was winging it since I didn't have a reference. Could you supply a good internet reference?

 

[rochfor1]

http://en.wikipedia.org/wiki/Complex_logarithm