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The power series \(1+\frac z{\Gamma(1+i)}+\frac{z^2}{\Gamma (1+2i)}+\frac{z^3}{\Gamma(1+3i)}+\cdots\) has a natural boundary! (La série entière \(1+\frac z{\Gamma(1+i)}+\frac{z^2}{\Gamma (1+2i)}+\frac{z^3}{\Gamma(1+3i)}+\cdots\) possède une frontière naturelle!) (French) Zbl 1236.30004

For each pair \(u\in\mathbb{C}^+:= \{z: \text{Re\,}z> 0\}\) and \(v\in\mathbb{R}^+:= \{x\in\mathbb{R}: x> 0\}\), the authors show that the non-lacunary power series \(\sum_{n\in\mathbb{Z}} \Gamma(u+ {2ivn\over\pi})\, z^n\) has \(\partial C_v\) as a natural boundary, where \(C_v:=\{z\in\mathbb{C}: e^{-|v|}< |z|< e^{|v|}\}\). The proof involves transforming this series into a lacunary Dirichlet series.

MSC:

30B10 Power series (including lacunary series) in one complex variable
30B30 Boundary behavior of power series in one complex variable; over-convergence
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