Page 458 - Electromagnetics

showing that F is independent of the angular variable θ. Expression (A.11) is termed
the Fourier–Bessel transform of f . The reader can easily verify that f can be recovered
from F through
∞

f (ρ, x 3 ,..., x N ) = F(p, x 3 ,..., x N )J 0 (ρp) pdp,
0
the inverse Fourier–Bessel transform.

A review of complexcontour integration

Some powerful techniques for the evaluation of integrals rest on complex variable the-
ory. In particular, the computation of the Fourier inversion integral is often aided by
these techniques. We therefore provide a brief review of this material. For a fuller
discussion the reader may refer to one of many widely available textbooks on complex
analysis.
We shall denote by f (z) a complex valued function of a complex variable z. That is,

f (z) = u(x, y) + jv(x, y),

where the real and imaginary parts u(x, y) and v(x, y) of f are each functions of the real
and imaginary parts x and y of z:
z = x + jy = Re(z) + j Im(z).

√
Here j = −1, as is mostly standard in the electrical engineering literature.

Limits, differentiation, and analyticity. Let w = f (z), and let z 0 = x 0 + jy 0 and
w 0 = u 0 + jv 0 be points in the complex z and w planes, respectively. We say that w 0 is
the limit of f (z) as z approaches z 0 , and write

lim f (z) = w 0 ,
z→z 0
if and only if both u(x, y) → u 0 and v(x, y) → v 0 as x → x 0 and y → y 0 independently.
The derivative of f (z) at a point z = z 0 is defined by the limit
f (z) − f (z 0 )

f (z 0 ) = lim ,
z→z 0 z − z 0
if it exists. Existence requires that the derivative be independent of direction of approach;
that is, f (z 0 ) cannot depend on the manner in which z → z 0 in the complex plane. (This

turns out to be a much stronger condition than simply requiring that the functions u and
v be differentiable with respect to the variables x and y.) We say that f (z) is analytic
at z 0 if it is differentiable at z 0 and at all points in some neighborhood of z 0 .
If f (z) is not analytic at z 0 but every neighborhood of z 0 contains a point at which
f (z) is analytic, then z 0 is called a singular point of f (z).

Laurent expansions and residues. Although Taylor series can be used to expand
complex functions around points of analyticity, we must often expand functions around
points z 0 at or near which the functions fail to be analytic. For this we use the Laurent

© 2001 by CRC Press LLC