Polya's representation (Boas Book)

Question

I am studying the Borel transform. If $f(z)$ is an entire function $f(z)=\sum_{n=0}^{\infty}a_nz^n$, then, the Borel transform is $F(z)=\sum_{n=0}^{\infty}\frac{n! a_n}{z^{n+1}}$.

Question: why $\int_{\Gamma}\left\{\sum_{n=0}^{\infty}n!a_n w^{-n-1}\sum_{k=0}^{\infty}\frac{z^kw^k}{k!}\right\}dw=2\pi i f(z)$?

I tried this with (Conway)

but i can't get the result.

Answer 1

I just solved it with $f(w)=w^k$ and nothing that $f^{(n)}(w)=k!$ if $n=k$