  In this paper we study the function chi(x1,x2,g) that determines the dressing
phase that appears in the all-loop Bethe Ansatz equations for the SL(2) sector
of N=4 super Yang-Mills theory. First, we consider the coefficients c_{r,s}(g)
of the expansion of chi(x1,x2,g) in inverse powers of x1, x2. We obtain an
expression in terms of a single integral valid for all values of the coupling
g. The expression is such that the small and large coupling expansion can be
simply computed in agreement with the expected results. This proves the, up to
now conjectured, equivalence of both expansions of the phase. The strong
coupling expansion is only asymptotic but we find an exact expression for the
value of the residue which can be seen to decrease exponentially with g.
  After that, we consider the function chi(x1,x2,g) itself and, using the same
method, expand it for small and large coupling. All small and large coupling
coefficients chi^(n)(x1,x2), for even and odd n, are explicitly given in terms
of finite sums or, alternatively, in terms of the residues of generating
functions at certain poles.

High Energy Physics - Theory