I have seen the definition of a principal logarithm as the logarithm whose imaginary part is in $(-\pi, \pi]$. However, I have also seen that the principal logarithm defined as the logarithm obtained from the branch cut that removes the negative real axis. The first allows one to take logarithms of negative numbers, while the second does not. Is one definition more standard than the other, or am I misinterpreting something?
Is the principal logarithm defined for negative numbers?
2
$\begingroup$
complex-analysis
-
0Give the principal branch of √(1 - z) – 2011-10-16
2 Answers
5
Branch cut is the curve of discontinuity of the function. It is not being removed from the plane.
The principal logarithm is discontinuous on the negative real axis with continuity from above, i.e.
$ \lim_{\epsilon \to 0^+} \log (-z + i \epsilon) = \log(-z) \qquad \lim_{\epsilon \to 0^+} \log (-z - i \epsilon) = \log(-z) - 2 \pi i $ for $z>0$.
4
The principal branch sometimes varies from author to author depending on their preference and what they are trying to accomplish. Removing the negative real axis gives essentially the same function, except it is defined on an open set, which is more useful for topics such as holomorphism.
A lot of questions in complex analysis deal only with open sets or open subsets.