Page 456 - Handbook Of Integral Equations

˜
For given f c (u), the function can be found by means of the Fourier cosine inversion formula

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

˜
The Fourier cosine transform (1) is denoted for brevity by f c (u)= F c f(x) . It follows from
2
formula (2) that the Fourier cosine transform has the property F = 1. There are tables of the
c
Fourier cosine transform (see Supplement 7) which prove useful in the solution of specific integral
equations.
Sometimes the asymmetric form of the Fourier cosine transform is applied, which is given by
the pair of formulas

∞ 2 ∞
ˇ
ˇ
f c (u)= f(x) cos(xu) dx, f(x)= f c (u) cos(xu) du. (3)
π
0 0

ˇ
The direct and inverse Fourier cosine transforms (3) are denoted by f c (u)= F c f(x) and f(x)=

–1 ˇ
F f c (u) , respectively.
c
7.5-2. The Fourier Sine Transform
Let a function f(x) be integrable on the semiaxis 0 ≤ x < ∞. The Fourier sine transform is defined
by

2 ∞
˜
f s (u)= f(x) sin(xu) dx, 0 < u < ∞. (4)
π
0
˜
For given f s (u), the function f(x) can be found by means of the inverse Fourier sine transform

2 ∞
˜
f(x)= f s (u) sin(xu) du, 0 < x < ∞. (5)
π 0

˜
The Fourier sine transform (4) is briefly denoted by f s (u)=F s f(x) . It follows from formula (5)
2
that the Fourier sine transform has the property F = 1. There are tables of the Fourier sine transform
s
(see Supplement 6), which are useful in solving specific integral equations.
Sometimes it is more convenient to apply the asymmetric form of the Fourier sine transform
defined by the following two formulas:

∞ 2 ∞
ˇ
ˇ
f s (u)= f(x) sin(xu) dx, f(x)= f s (u) sin(xu) du. (6)
0 π 0

ˇ
The direct and inverse Fourier sine transforms (6) are denoted by f s (u)= F s f(x) and f(x)=

–1 ˇ
F s f s (u) , respectively.
•
References for Section 7.5: V. A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), Yu. A. Brychkov and
A. P. Prudnikov (1989), W. H. Beyer (1991).
7.6. Other Integral Transforms
7.6-1. The Hankel Transform
The Hankel transform is defined as follows:

∞
˜
f ν (u)= xJ ν (ux)f(x) dx, 0 < u < ∞, (1)
0
1
where ν > – and J ν (x) is the Bessel function of the first kind of order ν (see Supplement 10).
2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 437