Page 459 - Electromagnetics

expansion, a generalization of the Taylor expansion involving both positive and negative
powers of z − z 0 :

∞ ∞ ∞

n a −n n
f (z) = a n (z − z 0 ) = + a n (z − z 0 ) .
(z − z 0 ) n
n=−∞ n=1 n=0
The numbers a n are the coefficients of the Laurent expansion of f (z) at point z = z 0 .
The first series on the right is the principal part of the Laurent expansion, and the second
series is the regular part. The regular part is an ordinary power series, hence it converges
in some disk |z−z 0 | < R where R ≥ 0. Putting ζ = 1/(z−z 0 ), the principal part becomes

n
∞ a −n ζ ; this power series converges for |ζ| <ρ where ρ ≥ 0, hence the principal part
n=1
converges for |z − z 0 | > 1/ρ r. When r < R, the Laurent expansion converges in the
annulus r < |z − z 0 | < R; when r > R, it diverges everywhere in the complex plane.
The function f (z) has an isolated singularity at point z 0 if f (z) is not analytic at z 0
but is analytic in the “punctured disk” 0 < |z − z 0 | < R for some R > 0. Isolated
singularities are classified by reference to the Laurent expansion. Three types can arise:

1. Removable singularity. The point z 0 is a removable singularity of f (z) if the principal
part of the Laurent expansion of f (z) about z 0 is identically zero (i.e., if a n = 0
for n =−1, −2, −3,...).
2. Pole of order k. The point z 0 is a pole of order k if the principal part of the Laurent
expansion about z 0 contains only finitely many terms that form a polynomial of
−1
degree k in (z − z 0 ) . A pole of order 1is called a simple pole.
3. Essential singularity. The point z 0 is an essential singularity of f (z) if the principal
part of the Laurent expansion of f (z) about z 0 contains infinitely many terms (i.e.,
if a −n = 0 for infinitely many n).

The coefficient a −1 in the Laurent expansion of f (z) about an isolated singular point z 0
is the residue of f (z) at z 0 . It can be shown that

1
a −1 = f (z) dz (A.12)
2π j
where is any simple closed curve oriented counterclockwise and containing in its interior
z 0 and no other singularity of f (z). Particularly useful to us is the formula for evaluation
of residues at pole singularities. If f (z) has a pole of order k at z = z 0 , then the residue
of f (z) at z 0 is given by

1 d k−1 k
a −1 = lim [(z − z 0 ) f (z)]. (A.13)
(k − 1)! z→z 0 dz k−1

Cauchy–Goursat and residue theorems. It can be shown that if f (z) is analytic
at all points on and within a simple closed contour C, then

f (z) dz = 0.
C
This central result is known as the Cauchy–Goursat theorem. We shall not offer a proof,
but shall proceed instead to derive a useful consequence known as the residue theorem.

© 2001 by CRC Press LLC