I met some problem when I dealt with the complex function $z^{\alpha}$($\alpha$ is not an integer). For example, $\alpha=1/2$. If $z\neq0$, $f(z)=z^{1/2}=\exp(1/2\log{z})$ which is holomorphic in $C\setminus{\{0\}}$. Also, if $|z|\leq1$ and $z\neq0$, we have $|f(z)|=|z|^{1/2}\leq1$. That is, $f$ is bounded on $\{z:|z|\leq1\}\setminus{\{0\}}$. By Riemann's theorem removable theorem (https://en.wikipedia.org/wiki/Removable_singularity#Riemann's_theorem), $z=0$ is a removable singularity. But it seems that it couldn't happen. Here is a contradiction. If above is true, the value $f(0)$ should be \begin{equation*} \begin{split} f(0) &=\frac{1}{2\pi{i}}\int_{|\zeta|=1}\frac{f(\zeta)}{\zeta-0}d\zeta\\ &=\frac{1}{2\pi{i}}\int_{0}^{2\pi}e^{\frac{i\theta}{2}}d\theta\\ &=2/\pi. \end{split} \end{equation*} Also, $f(0)$ should be \begin{equation*} \begin{split} f(0) &=\frac{1}{2\pi{i}}\int_{|\zeta|=1/2}\frac{f(\zeta)}{\zeta-0}d\zeta\\ &=\frac{1}{2\pi{i}}\int_{0}^{2\pi}\frac{\sqrt{2}}{2}e^{\frac{i\theta}{2}}d\theta\\ &=\sqrt{2}/\pi. \end{split} \end{equation*} This is a contradiction.
I don't know why this will happen. Actually, I thought $f(z)=z^{1/2}$ is not analytic at $z=0$. But I can't convince myself. Could you tell me something about the behavior of $f$ at $z=0$? Thank you very much.
It is not a removable singularity. Moreover, it is not an isolated singularity.
The classification into "removable, pole, essential" is for isolated singularities only: i.e., under the assumption that the function is holomorphic in a punctured neighborhood of the point. Riemann's theorem is also for this kind of singularities only.
The point $z=0$ is a branch point of the function $f(z) = z^{1/2}$, which is another story. It can be classified as an algebraic branch point, as for each $z$ in a neighborhood of $0$, there are only finitely many values taken by the branches of $f$. One can push some of the theory of Laurent series to such branch points by using series with fractional powers of $z$; the Complex Analysis book by Ahlfors does this.