Page 454 - Handbook Of Integral Equations

7.4. The Fourier Transform

7.4-1. Definition. The Inversion Formula
The Fourier transform is defined as follows:

∞
1
˜ –iux
f(u)= √ f(x)e dx. (1)
2π –∞
For brevity, we rewrite formula (1) as follows:
˜
˜
f(u)= F{f(x)}, or f(u)= F{f(x), u}.
˜
Given f(u), the function f(x) can be found by means of the inverse Fourier transform

1 ∞ iux
˜
f(x)= √ f(u)e du. (2)
2π –∞
Formula (2) holds for continuous functions. If f(x)hasa(finite) jump discontinuity at a point

1
x = x 0 , then the left-hand side of (2) is equal to f(x 0 – 0) + f(x 0 +0) at this point.
2
For brevity, we rewrite formula (2) as follows:
–1 ˜
–1 ˜
f(x)= F {f(u)}, or f(x)= F {f(u), x}.
7.4-2. An Asymmetric Form of the Transform

Sometimes it is more convenient to define the Fourier transform by

∞
ˇ
f(u)= f(x)e –iux dx. (3)
–∞
ˇ
ˇ
For brevity, we rewrite formula (3) as follows: f(u)= F{f(x)} or f(u)= F{f(x), u}.
In this case, the Fourier inversion formula reads

1 ∞ iux
ˇ
f(x)= f(u)e du, (4)
2π
–∞
–1 ˇ
and we use the following symbolic notation for relation (4): f(x)= F {f(u)},or f(x)=
–1 ˇ
F {f(u), x}.
7.4-3. The Alternative Fourier Transform
Sometimes, for instance, in the theory of boundary value problems, the alternative Fourier transform
is used (and called merely the Fourier transform) in the form

1 ∞ iux
F(u)= √ f(x)e dx. (5)
2π –∞
For brevity, we rewrite formula (5) as follows:

F(u)= F{f(x)}, or F(u)= F{f(x), u}.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 435