complex numberscomplex-analysis
Find a branch of $\log(z)$ on domain $\mathbb{C}\setminus T$ where $T=\{x+iy:x\ge 0, y=x^2\}$
I know the branch cut will be a parabola, branch cuts are usually rays, and my prof. did explain it briefly but could someone explain it in more detail? Greatly appreciated, thanks!
The way we define the Logarithm is the inverse of the exponential.
If $z= re^{i\theta}$,then $ln(z)=ln(r)+i\theta$.
Now , a problem that is there is, with this equation alone , logarithm is multi-valued i.e for every z there are infinitely many values for the logarithm . For $\theta$ differing by a value of $2\pi$ , one will get the same value. It isn't injective. This leads us restricting its domain so that each point(z) corresponds to only one value. This is referred to as a branch cut. Check the image from Wiki,
So, for log(z), simply remove any ray joining 0 and infinity.(Restricting complete rotation(0 to $2\pi$!). Also, see in the graph what happens. The principle log removes one particular ray, the negative real axis, I guess. This is only a convention, I guess.
Now, coming back to your question, you want log(iz)(PS:-The $iz$ only changes the labels on the axes) to be analytic in some given region. All you need to do is to remove a ray(not in the given region) to make it analytic in the given region.
A small circle around one of the zeros, the angle of $z^3+1$ increases by $2\pi$. The angle of its cube root increases by $2\pi/3$, so $f(x)$ changes by a factor $e^{2i\pi/3}$. So $f(x)$ is not well-defined, or at least not continuous, if that path is allowed.
Take a curved cut from $-1$, via $e^{i\pi/3}$, to $e^{-i\pi/3}$. Then any allowed path that goes around one root goes around all three roots of $z^3+1$. Each factor of $z^3+1$ increases in angle by $2\pi$, so $z^3+1$ increases in angle by $6\pi$. Its cube root increases in angle by $2\pi$, so $f(z)$ takes a consistent cube root of $z^3+1$.
Best Answer
Given a point $z=re^{i\theta}$, not on the parabola, with $0\le\theta\lt2\pi$, imagine starting at $(r,0)$ and going around the circle of radius $r$ counterclockwise until you get to $z$. If you get to $z$ without crossing the parabola, define $\log z=\log r+i\theta$. If you cross the parabola before you get to $z$, define $\log z=\log r+i(\theta-2\pi)$.