In this work n-armed Archimedian spiral wave solutions of the complex
Ginzburg-Landau equation are considered. These solutions are showed to depend
on two characteristic parameters, the so called twist parameter and the
asymptotic wavenumber. The existence and uniqueness of the value of the
asymptotic wavenumber, depending on the twist parameter, for which n-armed
Archimedian spiral wave solutions exist is a classical result, obtained back in
the 80s by Kopell and Howard. In this work we deal with a different problem,
that is, the asymptotic expression of the asymtptotic wavenumer for small
values of the twist parameter. Since the eighties, different heuristic
perturbation techniques, like formal asymptotic expansions, have conjectured an
asymptotic expression of which is exponentially small with respect to the twist
parameter. However, the validity of this expression has remained opened until
now, despite of the fact that it has been widely used for more than 40 years.
In this work, using a functional analysis approach, we finally prove the
validity of the asymptotic formula, providing a rigorous bound for its relative
error