Page 246 - Handbook Of Integral Equations

3.5-5. Kernels Containing a Combination of Trigonometric Functions

∞

25. cos(xt) + sin(xt) y(t) dt = f(x).
–∞
Solution:
1 ∞
y(x)= cos(xt) + sin(xt) f(t) dt.
2π
–∞
Up to constant factors, the function f(x) and the solution y(t) are the Hartley transform pair.
•
Reference: D. Zwillinger (1989).

∞

26. sin(xt) – xt cos(xt) y(t) dt = f(x).
0
3
This equation can be reduced to a special case of equation 3.7.1 with ν = .
2
Solution:
2 ∞ sin(xt) – xt cos(xt)
y(x)= f(t) dt.
2 2
π 0 x t
3.5-6. Equations Containing the Unknown Function of a Complicated Argument

π/2

27. y(ξ) dt = f(x), ξ = x sin t.
0
Schl¨ omilch equation.
Solution:
π/2
2
y(x)= f(0) + x f (ξ) dt , ξ = x sin t.

ξ
π
0
•
References: E. T. Whittaker and G. N. Watson (1958), F. D. Gakhov (1977).
π/2
k
28. y(ξ) dt = f(x), ξ = x sin t.
0
Generalized Schl¨ omilch equation.
This is a special case of equation 3.5.29 for n = 0 and m =0.
Solution:

x
2k k–1 d 1 k
y(x)= x k x k sin tf(ξ) dt , ξ = x sin t.
π dx 0
π/2
k
λ
29. sin ty(ξ) dt = f(x), ξ = x sin t.
0
This is a special case of equation 3.5.29 for m =0.
Solution:
2k k–λ–1 d λ+1 x λ+1 k
y(x)= x k x k sin tf(ξ) dt , ξ = x sin t.
π dx 0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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