Overview

An integral which is important in diffraction theory is

(C.1)

where ? is a positive real quantity, C is some contour in the complex ? = ?+ j ? plane, and

• f( ?) = u( ?, ?) + jv( ?, ?)

is an analytic function with non-zero imaginary part. Since ? is typically large, the exponential portion of the integrand is highly oscillatory and, to facilitate the evaluation of the integral, C is deformed into the path C _SDP shown in Fig. C-1. The new path is such that on C _SDP

(C.2)

where ? _s = ?s+ j ? _s is a point still to be determined and, because v is constant, the integrand is no longer oscillatory. From Cauchy's residue theorem we then have

(C.3)

where the residues are those of any poles which are captured, and I _b( ?) is the contribution from any branch cuts crossed in the deformation of C into C _SDP We will henceforth omit I _b( ?). The remaining integral in (C.3) is

(C.4)

and in view of (C.2)

(C.5)

Since ? is clearly real, (C.4) can be written as

(C.6)

and for ? ? 1 the exponential decreases rapidly away from ? = 0. The only significant contribution to the integral is from the immediate vicinity of ? _s, and we...