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138 Advanced Topics in Applied Mathematics
2
π
∞
0
2
π
a
a
2
+x
2
cosxξ dx = e
−aξ
, (3.230)
we obtain eigenfunctions
φ(x) =e
−ax
+
2
π
a
a
2
+x
2
. (3.231)
This eigenfunction corresponds to the eigenvalue of unity. We
could do this with any pair of functions and their transforms,
which illustrates the multiple eigenfunctions for our singular integral
equation. Also, by subtracting the transform from its function, we
obtain eigenvalues of −1.
2. Equations of Convolution Type
An integral equation of the form
1
√
2π
∞
−∞
k(x −t)u(t)dt = f (x), (3.232)
under the Fourier transform, becomes
K(ξ )U(ξ ) = F(ξ ). (3.233)
It may seem that we can solve for u from
U(ξ) =M(ξ )F (ξ), M(ξ ) =1/K. (3.234)
However, the reciprocals of Fourier transforms do not have inverses.
If K decays at infinity, M grows at infinity. If this growth is algebraic
(as opposed to exponential), we may divide M by a power of ξ, such as
ξ
n
, and compensate for this by multiplying F by ξ
n
. The factor ξ
−n
M
may have an inverse, and the inverse of ξ
n
F can be found if the nth
derivative of f exists.
As an illustration, consider the equation
1
√
2π
∞
−∞
u(t)dt
|x −t|
p
=f (x),0< p < 1. (3.235)