B.1 Numerical Wiener-Hopf Factorisation

A crucial step in the solution of Wiener-Hopf or dual integral equations is the factorisation (or splitting) of an even function F( ?) into a product of two functions such that

(B.1)

where ? = ? + j ?, F ₊( ?) is free of singularities and zeros in the upper ( ? > ? _-) half of the complex ?-plane, and F_( ?) has similar properties in the lower half ? > ? ₊, with ? ₊ > ? _-. For the factorisation to be possible, F( ?) must be analytic in the strip ? _- < ? < ? ₊ where ? may be vanishingly small. If we further demand that F( ?) ? 1 uniformly as ? ? ? in the strip, then (Mittra and Lee, 1971)

(B.2)

where

(B.3)

and C ₁ is the path shown in Fig. B-1. The last of the above conditions is not a restriction since we can always modify F( ?) to produce this behaviour. In addition, because F( ?) is an even function, we can set ? _- = - ? ₊, ensuring that the path C ₁ (for which ? = 0) remains within the strip as ? ₊ approaches zero. The integral (B.3) is not convenient...