Asymptotic expansions for Riemann-Hilbert problems

Let Gamma be a piecewise smooth contour in C, which could be unbounded and may have points of self-intersection. Let V (z, N) be a 2 x 2 matrix-valued function defined on Gamma, which depends on a parameter N. Consider a Riemann-Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Gamma with the jump matrix V (z, N). Assume that V (z, N) has an asymptotic expansion, as N-->infinity, on Gamma. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n-->infinity, for z. C\Gamma. Our method makes use of only complex analysis.