Page 642 - Handbook Of Integral Equations

12.4. Singular Integral Equations of the First Kind

12.4-1. The Simplest Equation With Cauchy Kernel
Consider the singular integral equation of the first kind

1 ϕ(τ)
dτ = f(t), (1)
πi L τ – t

where L is a closed contour. Let us construct the solution. In this relation we replace the variable t
1 dτ 1
by τ 1 , multiply by , integrate along the contour L, and change the order of integration
πi τ 1 – t
according to the Poincar´ e–Bertrand formula (see Subsection 12.2-6). Then we obtain

1 f(τ 1 ) 1 1 dτ 1
dτ 1 = ϕ(t)+ ϕ(τ) dτ . (2)
πi L τ 1 – t πi L πi L (τ 1 – t)(τ – τ 1 )

Let us calculate the second integral on the right-hand side of (2):

dτ 1 1 dτ 1 dτ 1 1
= – = (iπ – iπ)=0.
(τ 1 – t)(τ – τ 1 ) τ – t τ 1 – t τ 1 – τ τ – t
L L L
Thus,
1 f(τ)
ϕ(t)= dτ. (3)
πi L τ – t

The last formula gives the solution of the singular integral equation of the first kind (1) for a closed
contour L.

12.4-2. An Equation With Cauchy Kernel on the Real Axis
Consider the following singular integral equation of the first kind on the real axis:

1 ∞ ϕ(t)
dt = f(x), –∞ < x < ∞. (4)
πi t – x
–∞
Equation (4) is a special case of the characteristic integral equation on the real axis (see Subsec-
tion 13.2-4). In the class of functions vanishing at infinity, Eq. (4) has the solution

1 ∞ f(t)
ϕ(x)= dt, –∞ < x < ∞. (5)
πi t – x
–∞
–1
Denoting f(x)= F(x)i , we rewrite Eqs. (4) and (5) in the form

1 ∞ ϕ(t) 1 ∞ F(t)
dt = F(x), ϕ(x)= – dt. – ∞ < x < ∞. (6)
π t – x π t – x
–∞ –∞
The two formulas (6) are called the Hilbert transform pair (see Subsection 7.6-3).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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