  For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line
$(0,\infty)$ whose Hamiltonian $H$ is a.e. positive semidefinite, denote by
$q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that
the assignment $H\mapsto q_H$ is a bijection between trace-normalised
Hamiltonians and Nevanlinna functions. We prove that $q_H$ has an asymptotics
towards $i\infty$ whose leading coefficient is some (complex) multiple of a
regularly varying function if and only if the primitive $M$ of $H$ is regularly
or rapidly varying at $0$ and its off-diagonal entries do not oscillate too
much. The leading coefficient in the asymptotic of $q_H$ towards $i\infty$ is
related to the behaviour of $M$ towards $0$ by explicit formulae. The speed of
growth in absolute value depends only on the diagonal entries of $M$, while the
argument of the leading coefficient corresponds to the relative size of the
off-diagonal entries.
