I'm having problems finding the branch points of this complex function: $f(z)=z^i$.
I know how to find the branch points of the typical functions:
- $f(z)=z^p$, where $p$ is a rational non-integer. This has branch points at $z=0$ and $z=\infty$.
- $f(z)=\ln(z)$. This has two branch points, also at $z=0$ and $z=\infty$.
Now, for other complex functions that I have come across, the strategy was to convert it into one of those two forms, and work out from there. However, I am having trouble doing this for the function above, where the exponent is an imaginary number.
I have tried converting $z$ using the Euler identity and manipulating with it, but have not been able to get anywhere. Would appreciate some insights or hints on this problem. Thank you!
Edit:
So here's my attempt, not that it got me anywhere. \begin{align} \displaystyle f(z)&=\left(r\exp({i\theta})\right)^i \\ &= \left(r\exp({i\theta+2k\pi i}) \right)^i \\ &=r^i \exp(-\theta -2k\pi ) \end{align}