  The finite Hilbert transform T is a singular integral operator which maps the
Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By
extending the Parseval and Poincar\'e-Bertrand formulae to this setting, it is
possible to establish an inversion result needed for solving the airfoil
equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$
within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$
belonging to the union of all $L^p$ spaces with $p>1$. It is established (due
to a result of Stein) that $T$ cannot be extended to any domain space beyond
$LlogL$ whilst still taking its values in $L^1$, i.e., $T:LlogL\to L^1$ is
optimally defined.
