I'm confused about the way to approach this exercises, I've been introduced complex analysis a few weeks ago.
If I have, for example, $f(z)=\sqrt z$ what I'm tempted to do is to find some derivarives $f,f',f'',f''',...$ to see if I can guess the n-th term and write it's taylor polynomial around the origin.
Another way is to find some way to write $f(z)$ with terms whose series expansion are used frequently and well known.
My question is: what is necessary to be able to expand $f(z)$ as a power series around $z_0$? Is there a way am I missing? how can we do this kind of exercises?
Thanks for your time.
A Taylor series expansion is possible for a function $f\colon\mathbb C\to\mathbb C$ is possible (converges) at $z_0$ if $f$ is complex analytic in a neighborhood of the point $z_0$.
The square root function has two issues. First, at the origin it is not even analytic: it does not satisfy the Cauchy–Riemann equations because it is not even differentiable. In addition, outside the origin it is not single-valued. If you choose a branch (see below) of the square root, then it is analytic in a neighborhood of any $z_0\neq0$.
There is also a possibility to have a series expansion around singular points. This is known as a Laurent series, but at this stage you should focus on analytic functions and Taylor series. But even the Laurent series does not exist for all functions. It requires that the singularity is of a suitable nice kind, and the one of the square root at the origin does not qualify.
Every non-zero complex number has two square roots, just like a number $x>0$ has two square roots which differ by sign. Taking a branch means choosing one of the two in a consistent way so that the square root function becomes continuous and single-valued. This concept should appear at some later point in your studies in more detail, but this is the idea. Some other functions have more than two options to choose from. For example, the number $1+i$ has three cubic roots and infinitely many logarithms.