A UTD solution for an edge of a perfectly conducting wedge is presented which includes terms of order higher than the ordinary UTD. The problem is studied for three special cases: (i) plane and (ii) spherical wave incidence on a straight wedge with planar surfaces, and (iii) cylindrical wave incidence on a wedge with surfaces curved in the direction normal to the edge. This solution not only compensates the jump discontinuities in the GO field but also the discontinuities in the derivative. The solution found is exact for the special case of a plane wave incident on a halfplane. The solution with the higher order terms is found to be accurate when the large parameter is reduced by a factor of two as compared with the ordinary UTD solution.