Diffraction at a single ideally conducting slit

Maxwell's equations are solved when light, incident normally, is diffracted by a long single slit whose jaws are ideally conducting and of negligible thickness. Expressions, sums over Mathieu functions, are given for the E and B fields in the plane of the slit, and in the (Fraunhofer) far field, for both polarization possibilities. Graphs of these fields are presented, computed using Mathematica. Comparisons are made with the 'usual' optical assumptions (Kirchhoff boundary conditions, St Venant's hypothesis). The far field agrees excellently, as is well known from experiment; though agreement is best if the obliquity factor is set equal to 1. Some field components in the plane of the slit agree well with Kirchhoff, others less; we try to explain why these differences do not affect the far field. The diffracted field, and the total energy transmitted, become strongly polarization dependent for narrow slits (less than or similar to 0.4 lambda), and Kirchhoff then breaks down seriously.