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Figure 4.33: Integration contour used to evaluate the function F(x).

where k x0 = k cos φ 0 . Equations (4.418) and (4.419) comprise dual integral equations for
f . We may solve these using an approach called the Wiener–Hopf technique.
We begin by considering (4.419). If we close the integration contour in the upper
half-plane using a semicircle C R of radius R where R →∞, we find that the contribution
from the semicircle is
k x
lim f cos −1 e −|x|k xi e j|x|k xr dk x = 0
R→∞ k
C R
since x < 0. This assumes that f does not grow exponentially with R.Thus

−1 k x − jk x x
f cos e dk x = 0
k
C
where C now encloses the portion of the upper half-plane k xi > . By Morera’s theorem
[110],%citeLePage, the above relation holds if f is regular (contains no singularities
or branch points) in this portion of the upper half-plane. We shall assume this and
investigate the other properties of f that follow from (4.418).
In (4.418) we have an integral equated to an exponential function. To understand the
implications of the equality it is helpful to write the exponential function as an integral
as well. Consider the integral

∞+ j
1 h(k x ) 1 − jk x x
F(x) = e dk x .
2 jπ h(−k x0 ) k x + k x0
−∞+ j
Here h(k x ) is some function regular in the region k xi < , with h(k x ) → 0 as k x →∞.
If we choose so that −k xi > > −k xi cos θ 0 and close the contour with a semicircle in
the lower half-plane (Figure 4.33), then the contribution from the semicircle vanishes for
large radius and thus, by Cauchy’s residue theorem, F(x) =−e jk x0 x . Using this (4.418)
can be written as
∞+ j

f cos h(k x ) 1
−1 k x ˜
k E 0 − jk x x
− e dk x = 0.
2 2
k − k x 2 jπ h(−k x0 ) k x + k x0
−∞+ j
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