Let $f:\mathbb{R\to R}$ be a smooth function and suppose that it has a unique zero at $x=a$. For every $z\in\mathbb{C}$, define a function $f_z(x)=\left|\frac{f(x)}{x-a}\right|^z$. Find an expansion for $f_z(x)$.
If $z$ was real I could put it inside the absolute value and try to find a Taylor series for $f_z(x)$. However, since $z$ can have a non-zero imaginary part it's not possible and in fact $f_z(x)$ isn't analytic. Can I find an explicit expansion for $f_z(x)$?
EDIT: In fact, can we think on $f_z$ as a function which is analytic in $|\cdot|^z$ and expand it to $\sum_{n=0}^\infty a_n|x-a|^{nz}$?