Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form $$\sum_{n=0}^\infty a_nz^{-n}$$ as $z\to\infty$ (in some sector)?
If so, how can it be computed?
Thank you all very much in advance!
Regards,
Frank
No. That would require $z^{n-x}\to a_n$ in this sector, where $a_nz^{-n}$ is the first nonzero term in the sum. Taking absolute values, you can see that this will not happen unless $x$ is an integer.