1. Help with branch cuts

Q: How do I algebraically find the appropriate branch cut of a function? For example, lets consider $f(z)=log(z^{2})$.

If we choose to use the principle branch of the log function then $f'(z)=\frac{2}{z}$ which is undefined at $0$. I know that much, but the text I am using says $log(z^{2})$ is analytic on the set $A=\{z:z\neq\\0$ and $arg(z)\neq\\+\frac{\pi}{2}$ or $-\frac{\pi}{2}\}$. I am not seeing why it is this set.

In general, is there a way to assume nothing about your function other than maybe the log is defined on the principle branch and solve for the appropriate branch cuts? I can find the branch points, but not sure how to construct the proper cuts.

Thanks

2. Originally Posted by Danneedshelp
Q: How do I algebraically find the appropriate branch cut of a function? For example, lets consider $f(z)=log(z^{2})$.

If we choose to use the principle branch of the log function then $f'(z)=\frac{2}{z}$ which is undefined at $0$. I know that much, but the text I am using says $log(z^{2})$ is analytic on the set $A=\{z:z\neq\\0$ and $arg(z)\neq\\+\frac{\pi}{2}$ or $-\frac{\pi}{2}\}$. I am not seeing why it is this set.

In general, is there a way to assume nothing about your function other than maybe the log is defined on the principle branch and solve for the appropriate branch cuts? I can find the branch points, but not sure how to construct the proper cuts.

Thanks
Although convention might dictate a particular branch cut, in reality many branch cuts are possible. To define a branch cut, you first need to know where the branch point(s) is(are).