Conformal mapping of o-minimal corners (Q2907399)

This paper is in the spirit of finding consequences of o-minimality in expansions of the field of real numbers, and is a continuation of [\textit{T. Kaiser}, Proc. Lond. Math. Soc. (3) 98, No. 2, 427--444 (2009; Zbl 1168.03027)]. If a simply connected domain in the plane is definable in a polynomially bounded o-minimal structure, then at a corner point \(0\) a biholomorphism from the upper half-plane behaves like \(z^{\alpha}\), where \(\alpha\pi\) is the measure of the angle at the corner (two conditions of [\textit{S. Warschawski}, Math. Z. 35, 321--456 (1932; Zbl 0004.40401; JFM 58.0359.01)] are satisfied). In the special case of the structure \(\mathbb{R}^{\mathbb{R}}_{\mathrm{an}}\) (Main Theorem 2.9) an infinite expansion containing logarithmic terms (of Lehman type) is given. Here, rationality of \(\alpha\) is irrelevant. A result of Gaier about the derivatives of the biholomorphic map is also extended to \(\mathbb{R}^{\mathbb{R}}_{\mathrm{an}}\).