Page 455 - Handbook Of Integral Equations

For given F(u), the function f(x) can be found by means of the inverse transform

1 ∞ –iux
f(x)= √ F(u)e du. (6)
2π –∞
For brevity, we rewrite formula (6) as follows:

–1
–1
f(x)= F {F(u)}, or f(x)= F {F(u), x}.
The function F(u) is also called the Fourier integral of f(x).
We can introduce an asymmetric form for the alternative Fourier transform similarly to that of
the Fourier transform:

∞ 1 ∞
ˇ
ˇ
F(u)= f(x)e iux dx, f(x)= F(u)e –iux du, (7)
2π
–∞ –∞

ˇ
ˇ
where the direct and the inverse transforms (7) are briefly denoted by F(u)= F f(x) and f(x)=

ˇ
ˇ –1 ˇ
ˇ –1 ˇ
ˇ
F F(u) ,orby F(u)= F f(x), u and f(x)= F F(u) x .
7.4-4. The Convolution Theorem for the Fourier Transform
The convolution of two functions f(x) and g(x)isdefined as

1 ∞
f(x) ∗ g(x) ≡ √ f(x – t)g(t) dt.
2π –∞
By performing substitution x – t = u, we see that the convolution is symmetric with respect to the
convolved functions: f(x) ∗ g(x)= g(x) ∗ f(x).
The convolution theorem states that

F f(x) ∗ g(x) = F f(x) F g(x) . (8)
For the alternative Fourier transform, the convolution theorem reads

F f(x) ∗ g(x) = F f(x) F g(x) . (9)

Formulas (8) and (9) will be used in Chapters 10 and 11 for solving linear integral equations
with difference kernel.

•
References for Section 7.4: V. A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), Yu. A. Brychkov
and A. P. Prudnikov (1989), W. H. Beyer (1991).

7.5. The Fourier Sine and Cosine Transforms

7.5-1. The Fourier Cosine Transform

Let a function f(x) be integrable on the semiaxis 0 ≤ x < ∞. The Fourier cosine transform is defined
by

2 ∞
˜
f c (u)= f(x) cos(xu) dx, 0 < u < ∞. (1)
π
0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 436