Laurent series from entire functions (Q2725489)

Let \(r>0\), \(z_0 \in \mathbb{C}\), and let \(f\) be an entire function such that the power series \(\sum_{j=0}^\infty f^{(j)}(z_0)z^j\) converges for \(|z|<r\). Such a function is necessarily of exponential type, and an example is given by \(f(z)=e^{z/r}\). For \(j \in \mathbb{N}\) let \(A^jf\) denote the entire function which satisfies \((A^jf)^{(j)}=f\) and \((A^jf)(0)=0\). Then the Laurent series NEWLINE\[NEWLINE \phi(z) = \sum_{j=1}^\infty (A^jf)(z_0)z^{-j} + \sum_{j=0}^\infty f^{(j)}(z_0)z^j \tag{1} NEWLINE\]NEWLINE defines a function \(\phi\) which is holomorphic in \(0<|z|<r\). The main purpose of this article is to characterize those functions \(\phi\) holomorphic in \(0<|z|<r\) which are of the form (1) for some entire function \(f\). NEWLINENEWLINENEWLINELet \(r>0\) and \(z_0 \in \mathbb{C}\) be given. If \(\phi\) is a holomorphic function in \(0<|z|<r\), then \(\phi\) can be represented in the form (1) for some entire function \(f\) if and only if the limit \(\lim_{z \to 0}{[e^{-z_0/z}\phi(z)]}\) exists. In this case, \(f\) is uniquely determined, and \(z_0\) is the only point for which this limit exists. The proof is fairly elementary and uses only well-known facts from complex function theory.