Page 457 - Handbook Of Integral Equations

˜
For given f ν (u), the function f(x) can be found by means of the Hankel inversion formula

∞
˜
f(x)= uJ ν (ux)f ν (u) du, 0 < x < ∞. (2)
0
α
β
Note that if f(x)= O(x )as x → 0, where α + ν + 2 > 0, and f(x)= O(x )as x →∞, where
β + 3 < 0, then the integral (1) is convergent.
2
The inversion formula (2) holds for continuous functions. If f(x)hasa(finite) jump discontinuity
1
at a point 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 denote the Hankel transform (1) by f ν (u)= H ν f(x) . It follows from
2
formula (2) that the Hankel transform has the property H =1.
ν
7.6-2. The Meijer Transform
The Meijer transform is defined as follows:

2 ∞ √
ˆ
f µ (s)= sxK µ (sx)f(x) dx, 0 < s < ∞, (3)
π
0
where K µ (x) is the modified Bessel function of the second kind (the Macdonald function) of order µ
(see Supplement 10).
˜
For given f µ (s), the function f(x) can be found by means of the Meijer inversion formula
c+i∞
1 √
ˆ
f(x)= √ sxI µ (sx)f µ (s) ds, 0 < x < ∞, (4)
i 2π c–i∞
where I µ (x) is the modified Bessel function of the first kind of order µ (see Supplement 10). For
the Meijer transform, a convolution is defined and an operational calculus is developed.

7.6-3. The Kontorovich–Lebedev Transform and Other Transforms

The Kontorovich–Lebedev transform is introduced as follows:

∞
F(τ)= K iτ (x)f(x) dx, 0 < τ < ∞, (5)
0
where K µ (x) is the modified Bessel function of the second kind (the Macdonald function) of order µ
√
(see Supplement 10) and i = –1.
For given F(τ), the function can be found by means of the Kontorovich–Lebedev inversion
formula
∞
2
f(x)= τ sinh(πτ)K iτ (x)F(τ) dτ, 0 < x < ∞. (6)
2
π x 0
There are also other integral transforms, of which the most important are listed in Table 3 (for the
constraints imposed on the functions and parameters occurring in the integrand, see the references
given at the end of this section).

© 1998 by CRC Press LLC

Page 438
© 1998 by CRC Press LLC