Branch Points of the function $z^{(1 + i)}$

Question

I don't seem to understand what branch points and branch cuts are. Can anyone tell me the concepts by finding the branch points and the branch cuts of the following funcition. $$z^{(1 + i)}$$

Answer 1

Take any $z,a\in\Bbb{C}.$ Then usually $z^a$ is defined as $e^{a\log z},$ where $\log z=\ln |z|+\arg z$ is the complex logarithm of $z.$ Therefore $$z^{1+i}=e^{(1+i)\ln|z|+(-1+i)\arg(z)}.$$ Obviously, once we travel along a closed path enclosing the origin in the counterclockwise direction $$\arg(z)\to 2\pi+\arg(z).$$
Hence $z=0$ is a branch point of $z^{1+i}.$
Also changing the variable $z\to\dfrac1z,$ we can see that there is another branch point at $z=\infty.$
These are the only branch points of this function.