10.8.2 Series with complex terms

Problem: Find all expansions of the given function \( f(z) \) in powers of \( z-a \) and indicate the domains of these expansions. Remark 1. For a multivalued function \( \sqrt[3]{z} \mathrm{p} \) we consider the branch, which that takes real values on the positive part of the real axis. Remark 2. For the multivalued function \( \arctan z \) we consider the branch that takes real values on the positive part of the real axis. In this case, there is a representation: \[ \begin{array}{l} \arctan z=\int_{0}^{z} \frac{d z}{1+z^{2}}=\frac{\pi}{2}+\int_{\infty}^{z} \frac{d z}{1+z^{2}} \\ f(z)=\frac{z-1}{\sqrt[3]{z^{3}-3 z^{2}+3 z}} \text { in powers of }(z-1) . \end{array} \]