On the Theory of Complex Rays

The article surveys the application of complex ray theory to the scalar Helmholtz equation in two dimensions. The rst objective is to motivate a framework within which complex rays may be used to make predictions about waveeelds in a wide variety of geometrical conngurations. A crucial ingredient in this framework is the r^ ole played by Stokes' phenomenon in determining the regions of existence of complex rays. The identiication of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterisations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex wkb expansion of these waveeelds. Examples will be given for several cases of physical importance.