Branch of logarithm which is real when z>0 I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook: "For complex z!=0,log(z) denotes that branch of the logarithm which is real when z>0.What does this mean?
Amiya Melendez
Answered question
2022-10-13
Branch of logarithm which is real when z>0 I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook: "For complex denotes that branch of the logarithm which is real when What does this mean?
Answer & Explanation
Carly Yang
Beginner2022-10-14Added 19 answers
It doesn't mean anything unless you specify the region you want to define the complex logarithm on. The statement seems to try to do this with , but that is impossible: In general, the complex logarithm is only defined up to , , and if your winds around , you cannot make the function continuous. Instead, could be any simply connected subset of , and the usual choice is to remove the nonpositive reals from , i.e. . With this (or any other simply connected that contains the positive reals), it is possible to pick the branch that is real (i.e. is the well-known real logarithm) for positive reals.
Maribel Mcintyre
Beginner2022-10-15Added 3 answers
Let be the principal branch of the logarithm, that corresponds to a cut along the negative real numbers. Now, consider any branch of that is defined on a connected neighborhood of . Clearly, is a continuous function on that satisfies so, takes its values in , and since it is continuous, it must be constant. Thus, there is such that and this happens on the largest connected open set that contains , and on which both functions are analytic (by analytic continuation). Thus, the statement says that we choose the logarithmic function the corresponds to because otherwise would not be real for real