Page 250 - Handbook Of Integral Equations

3.7-2. Kernels Containing Modified Bessel Functions

b

4. I ν (λx)– I ν (λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x)= I ν (λx), where I ν is the modified Bessel
function of the first kind.
b

5. K ν (λx)– K ν (λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x)= K ν (λx), where K ν is the modified Bessel
function of the second kind (the Macdonald function).

∞ √

6. zt K ν (zt)y(t) dt = f(z).
0
Here K ν is the modified Bessel function of the second kind.
Up to a constant factor, the left-hand side of this equation is the Meijer transform of y(t)
(z is treated as a complex variable).
Solution:
1 c+i∞ √
y(t)= zt I ν (zt)f(z) dz.
πi
c–i∞
For specific f(z), one may use tables of Meijer integral transforms to calculate the integral.
•
Reference: V. A. Ditkin and A. P. Prudnikov (1965).

∞

7. K 0 |x – t| y(t) dt = f(x).
–∞
Here K 0 is the modified Bessel function of the second kind.
Solution:
2
∞
1 d
y(x)= – – 1 K 0 |x – t| f(t) dt.
π 2 dx 2
–∞
•
Reference: D. Naylor (1986).

3.7-3. Other Kernels
√
a 2 xt
y(t) dt
8. K = f(x).
0 x + t x + t
1
dt
Here K(z)= is the complete elliptic integral of the first kind.
2 2
2
0 (1 – t )(1 – z t )
Solution:
a t

4 d tF(t) dt d sf(s) ds
y(x)= – √ , F(t)= √ .
2
π dx x t – x 2 dt 0 t – s 2
2
2
•
Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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