Transition functions for high-frequency diffraction by a curved perfectly conducting wedge. I. Canonical solution for a curved sheet

Starting from the canonical Hilbert solution for a perfectly conducting, cylindrically curved sheet of semi-infinite angular extent illuminated by a line source, transition functions are derived for illumination and/or observation directions grazing the sheet surface (or its geometrical continuation) near the edge. This is done by using the Langer-Olver asymptotic formulas for certain Hankel and Bessel function factors in the Hilbert solution. Both the soft (transverse magnetic) and the hard (transverse electric) cases are considered. Depending on the positions of the source and/or the observer, the resulting functions include the Fock functions, the incomplete Airy function and its derivative, and a pair of new canonical functions. The results obtained can serve as a check for heuristic solutions for a curved wedge