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where the operator R 1 is defined by formula (41) of Subsection 13.2-4.
In the expression for the second summand on the left-hand side in (45), the operation R 1 with
respect to the variable t commutes with the operation of integration with respect to τ. Therefore,
Eq. (45) can be rewritten in the form

t

ϕ(t)+ R K(t, τ) ϕ(τ) dτ = R 1 [f(t)] + b(t)Z(t)P ν–p–1 (t), (46)
1
L
t
where the superscript t at the symbol of the operator R means that the operation is performed with
1
respect to the variable t.
Since the operator R 1 is bounded, it follows that the resulting integral equation (46) is a Fredholm
equation, and hence the regularization problem for the singular equation (44) is solved.
It follows from the general theory of the regularization that Eq. (44) is equivalent to Eq. (46) for
ν – p ≥ 0 and to Eq. (46) and a system of functional equations for ν – p <0.
In conclusion we note that for the above cases of singular integral equations, the Fredholm
theorems fail in general.
Remark 3. Exceptional cases of singular integral equations with Cauchy kernel can be reduced
to equations of the normal type.

13.4-8. The Complete Equation With Hilbert Kernel
Consider the complete singular integral equation with Hilbert kernel (see Subsection 13.1-2)

b(x) 2π ξ – x
2π
a(x)ϕ(x) – cot ϕ(ξ) dξ + K(x, ξ)ϕ(ξ) dξ = f(x). (47)
2π 0 2 0

Let us show that Eq. (47) can be reduced to a complete singular integral equation with a kernel
of the Cauchy type, and in this connection, the theory of the latter equation can be directly extended
to Eq. (47). Since the regular parts of these two types of equations have the same character, it
follows that it suffices to apply the relationship between the Hilbert kernel and the Cauchy kernel
(see Subsection 12.4-5):

dτ 1 ξ – x i
= cot dξ + dξ. (48)
τ – t 2 2 2
Hence,

1 ξ – x dτ 1 dτ
cot dξ = – , (49)
2 2 τ – t 2 τ
where t = e ix and τ = e iξ are the complex coordinates of points of the contour L, that is, the unit
circle.
On replacing the Hilbert kernel in Eq. (47) with the expression (49) and on substituting x = –i ln t,
ξ = –i ln τ, and dξ = –iτ –1 dτ, after obvious manipulations we reduce Eq. (47) to a complete singular
integral equation with Cauchy kernel of the form

ib 1 (t) ϕ 1 (τ)
a 1 (t)ϕ 1 (t) – dτ + K 1 (t, τ) dτ = f 1 (t). (50)
πi L τ – t L
The coefficient of the Riemann problem corresponding to Eq. (50) is

a 1 (t)+ ib 1 (t) a(x)+ ib(x)
D(t)= = , (51)
a 1 (t) – ib 1 (t) a(x) – ib(x)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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