Analytic continuation via calculus of residues (Q268057)

The author uses the calculus of residues to give a proof of the following theorem: Let \(R > 0\). Let \(f\) be a complex-valued function holomorphic in an open set containing the closed disc of radius \(R\) centered at the real point \(R\). Moreover, let \(f\) be holomorphic in a small open disc around the origin where it has the expansion \[ f(z) = \sum\limits_{n = 0}^\infty a_n z^n. \] Then for any \(w\) in the open disc of radius \(R\) centered at \(R\) the formula \[ f(w) = \lim_{a \to \infty} \sum_{k = 0}^\infty \gamma\left(k + 1, \frac{a R}{w}\right) \frac{a_k}{k!} w^k \] holds, where \[ \gamma(\alpha, z) = \mathrm{e}^{-z} \sum\limits_{k = 0}^\infty \frac{z^{\alpha + k}}{\prod_{j = 0}^k (\alpha + j)} \] is the incomplete gamma function with parameter \(\alpha\).