Page 461 - Electromagnetics

Figure A.2: Derivation of the contour deformation principle.

Principal value integrals. We must occasionally carry out integrations of the form
∞

I = f (x) dx
−∞
where f (x) has a finite number of singularities x k (k = 1,..., n) along the real axis. Such
singularities in the integrand force us to interpret I as an improper integral. With just
one singularity present at point x 1 , for instance, we define

∞ ∞
x 1 −ε
f (x) dx = lim f (x) dx + lim f (x) dx
ε→0 η→0
−∞ −∞ x 1 +η
provided that both limits exist. When both limits do not exist, we may still be able to
obtain a well-defined result by computing
x 1 −ε ∞

lim f (x) dx + f (x) dx
ε→0
−∞ x 1 +ε
(i.e., by taking η = ε so that the limits are “symmetric”). This quantity is called the
Cauchy principal value of I and is denoted

∞

P.V. f (x) dx.
−∞
More generally, we have

∞
x 1 −ε x 2 −ε
P.V. f (x) dx = lim f (x) dx + f (x) dx +
ε→0
−∞ −∞ x 1 +ε
x n −ε ∞
+ ··· + f (x) dx + f (x) dx
x n−1 +ε x n +ε
for n singularities x 1 < ··· < x n .
In a large class of problems f (z) (i.e., f (x) with x replaced by the complex variable
z) is analytic everywhere except for the presence of finitely many simple poles. Some
of these may lie on the real axis (at points x 1 < ·· · < x n , say), and some may not.
Consider now the integration contour C shown in Figure A.3. We choose R so large and
ε so small that C encloses all the poles of f that lie in the upper half of the complex

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