Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$ where $\gamma_{z_0}$ is a small circle around $z_0$. By expanding the exponential, I find that $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz=\sum_{n=0}^\infty \frac{1}{(n+1)!}\oint_{\gamma_{z_0}} \frac{f(z)}{(z-z_0)^{n+1}}dz.$$ The integral can be evaluated using Cauchy's integral formula, $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz=2\pi i\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!(n+1)!},$$ with $f^{(n)}$ the $n$-the derivative of $f$.
It seems that the result could be related more directly to $f$, having similarities with the Taylor series $f$. To be precise, if the $(n+1)!$ was removed, the result would be $2 \pi i f(z_0+1)$. I was wondering
a) if such Borel-like sums have been studied for Taylor series of holomorphic functions, and
b) if there is a more direct relation between the period integral and $f$ that would allow to compute the sum.
Many thanks for sharing your ideas!