On the solution of the Helmholtz equation on regions with corners

The solution of elliptic partial differential equations on regions with corners is a famously refractory problem. The solutions are known to be singular at corners, and one of the major difficulties has been finding a precise description of their behavior. In this talk, I observe that when the Helmholtz equation is solved using integral equations of classical potential theory, the solutions are explicitly representable by certain series of known singular functions (in particular, Bessel functions of noninteger order). These explicit representations lead to highly accurate and efficient numerical algorithms for the solution of the Helmholtz equation on domains with corners.