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NEOCLASSICAL THEORY OF INTERACTION 105

From these observations, the real and imaginary part of impedance () must be related as

1 ∞ ()
() = − ∫
−∞ − (2.110)
�
1 ∞ ()
() = ∫
−∞ −
In case of transmission or reflection coefficient, it is convenient first to separate the magnitude
and phase taking the natural logarithm of () or () and only then to apply the K-K
relations
ln�()� = ln(|()|) + ()
⎫
⎪
1 ∞ ln(|()|)
() = − ∫ (2.111)
−∞ −
⎬
1 ∞ ()
ln(|()|) = ∫ ⎪
−∞ − ⎭
Step 3 is the transform of the integrals in (2.109) making them suitable for numerical evaluation
by subtracting out the integrand divergence at the point = . Taking into account that
1 ∞ 1 +∞ = 0 (2.112)
∫ = (ln| − |)|
−∞ − −∞
we can rewrite (2.109) as

′
′
1 ∞ ()− ()
′′
() = − ∫

−∞ − � (2.113)
′′
′′
1
′
() − 1 = ∫ ∞ ()− ()

−∞ −
L’Hopital’s rule and the fact that () and () are indefinitely differentiable functions give
′′
′

us assurance that the ratio in each of integrands does not diverge any more at = .
2.10.2 Eliminating Negative Frequencies in the K-K Relations
The KK relations in (2.113) spin positive and negative frequencies. It follows from (2.106) that
the real part of () is an even while its imaginary part is an odd function of frequency

′
′
(−) = ()
� (2.114)
′′
′′
(−) = − ()

Step 4. Due to (2.114) the equation (2.113) can be transformed into
′ ′
()− ()
∞
2
′′
() = − ∫

0 − + � (2.115)
′′
′′
2 ∞ ()− ()
′
() = 1 + ∫

0 − +
2.10.3 Remote Sensing and K-K Relations
One of the most remarkable applications of the K-K relations is the remote sensing aka
acquisition of information about some object or objects without being in physical contact with
them. Particularly, the critical mission of such sensing is the identification of remote
object/objects. We are doing such identification at a distance every moment of our life using
our eyes, ears, nose as a smell sensor and quite sophisticated recognition algorithms buried deep
in our brain cells. If natural human sensors are not worldly-wise the required for identification
information can be collected by all kind of active and passive radars, microphones and cameras,
spectrometers and radiometers, microscopes and telescopes, multiple devices for magnetic