From Stein's complex analysis: prove that if we take $$ f(z)=\frac{1}{(1-z)^\alpha} \quad\text{for }|z|\lt1 $$ (defined in terms of the principal branch of the logarithm) where $\alpha$ is a fixed complex number, then $$ f(z)=\sum_{n=0}^\infty a_n(\alpha) z^n $$ with $$ a_n(\alpha)\sim\frac{1}{\Gamma(\alpha)}n^{\alpha-1}\quad\text{as}\;n\to\infty. $$
:added
i have calculated $$ a_n(\alpha)=\frac{\Gamma(n+\alpha)}{n!\Gamma(\alpha)} $$ so it remains to show: $$ \frac{\Gamma(n+\alpha)}{\Gamma(n+1)}\sim n^{\alpha-1}\text{for large n} $$ but i dont see how to prove it
Depending on how one chooses to define the Gamma function this may be a direct result of the definition. One such way to define the Gamma function is as the unique function $\Gamma$ satisfying:
1) $\Gamma(1)=1$.
2) $\Gamma(z+1)=z\Gamma(z)$.
3) $\Gamma(z+\alpha)\sim z^\alpha\Gamma(z)$.
which can be shown to be equivalent to the limit definition. Your question follows directly from the last condition above.