Source: Notre Dame J. Formal Logic Volume 53, Number 1
(2012), 67-77.
We show that the ordered field of real numbers with restricted
$\mathbb{R}_{\mathscr{H}}$-definable analytic functions admits quantifier
elimination if we add a function symbol $^{-1}$ for the function $x\mapsto
\frac{1}{x}$ (with $0^{-1}=0$ by convention), where $\mathbb{R}_{\mathscr{H}}$
is the real field augmented by the functions in the family $\mathscr{H}$ of
restricted parts (real and imaginary) of holomorphic functions which satisfies
certain conditions. Further, with another condition on $\mathscr{H}$ we show
that the structure ($\mathbb{R}_{\mathscr{H}}$, constants) is strongly model
complete.
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