Page 455 - Electromagnetics

The first of these is particularly useful when a problem has been solved in the
frequency domain and the solution is found to be a product of two or more functions
of k.
13. Parseval’s identity. We have

∞ 1 ∞
2 2
| f (x)| dx = |F(k)| dk.
2π
−∞ −∞
Computations of energy in the time and frequency domains always give the same
result.
14. Differentiation. We have
n
n
d f (x) n n d F(k)
↔ ( jk) F(k) and (− jx) f (x) ↔ .
dx n dk n
The Fourier transform can convert a differential equation in the x domain into an
algebraic equation in the k domain, and vice versa.
15. Integration. We have
x
F(k)
f (u) du ↔ π F(k)δ(k) +
jk
−∞
where δ(k) is the Dirac delta or unit impulse.

Generalized Fourier transforms and distributions. It is worth noting that many
useful functions are not Fourier transformable in the sense given above. An example is
the signum function

−1, x < 0,
sgn(x) =
1, x > 0.
Although this function lacks a Fourier transform in the usual sense, for practical purposes
it may still be safely associated with what is known as a generalized Fourier transform.A
treatment of this notion would be out of place here; however, the reader should certainly
be prepared to encounter an entry such as

sgn(x) ↔ 2/jk
in a standard Fourier transform table. Other functions can be regarded as possessing
transforms when generalized functions are permitted into the discussion. An important
example of a generalized function is the Dirac delta δ(x), which has enormous value
in describing distributions that are very thin, such as the charge layers often found
on conductor surfaces. We shall not delve into the intricacies of distribution theory.
However, we can hardly avoid dealing with generalized functions; to see this we need
look no further than the simple function cos k 0 x with its transform pair

cos k 0 x ↔ π[δ(k + k 0 ) + δ(k − k 0 )].

The reader of this book must therefore know the standard facts about δ(x): that it
acquires meaning only as part of an integrand, and that it satisfies the sifting property

∞

δ(x − x 0 ) f (x) dx = f (x 0 )
−∞

© 2001 by CRC Press LLC