Given any fixed $u\in\mathbb{C}$, does there exist a general form of the $n$-th term of a power series of $z^u$ ?
For such $u$ we would have a Laurent Series expansion like $$c_n=\frac{1}{2\pi i}\oint z^u\frac{dz}{z^{1+n}}$$ and $$z^u=\sum_{n\in\mathbb{Z}} c_nz^n$$ convergent in some open annulus/disk/subset of the complex plane.
Is there any simplified general formula for $c_n=c_n(u)$ in terms of elementary/special functions, even in fractional powers such as Puiseux series?
As you can read already in the comments, $z\mapsto z^u$ is in general not a proper function since the $\log$ in the definition is not jet a unique value. Note that for $u\in\mathbb{Z}_{<0}$ this multivalued $\log$ is just deleted again by the $\exp$ function's periodicity. So these $u$, which may have made you writing this question, are a very special case to still yield holomorphic functions around $0$, whose Laurent series then consequently (makes sense and) converge.