Page 659 - Handbook Of Integral Equations

Just as for the case of a finite contour, the characteristic integral equation

b(x) ∞ ϕ(τ)
a(x)ϕ(x)+ dτ = f(x) (25)
πi τ – x
–∞
can be reduced by means of the Cauchy type integral

1 ∞ ϕ(τ)
Φ(z)= dτ (26)
2πi τ – z
–∞
and the Sokhotski–Plemelj formulas (see Subsection 12.2-5), to the following Riemann boundary
value problem for the real axis (see Subsection 12.3-8):

a(x) – b(x) – f(x)
+
Φ (x)= Φ (x)+ , –∞ < x < ∞. (27)
a(x)+ b(x) a(x)+ b(x)
We assume that
2
2
a (x) – b (x) = 1, (28)

2
2
because Eq. (25) can always be reduced to case (28) by the division by a (t) – b (t). Note that the
index ν of the integral equation (25) is given by the formula
a(x) – b(x)
ν = Ind . (29)
a(x)+ b(x)
In this case for ν ≥ 0 we obtain

b(x)Z(x) ∞ f(τ) dτ P ν–1 (x)
ϕ(x)= a(x)f(x) – + b(x)Z(x) , (30)
πi Z(τ) τ – x (x + i) ν
–∞
where
–ν/2
x – i
+
–
Z(x)=[a(x)+ b(x)]X (x)=[a(x) – b(x)]X (x)= e G(x) ,
x + i
1 ∞ τ – i
–ν a(τ) – b(τ) dτ
G(x)= ln .
2πi τ + i a(τ)+ b(τ) τ – x
–∞
For the case in which ν ≤ 0 we must set P ν–1 (x) ≡ 0. For ν < 0, we must also impose the solvability
conditions
∞

f(x) dx
=0, k =1, 2, ... , –ν. (31)
Z(x) (x + i) k
–∞
For the solution of Eq. (25) in the class of functions bounded at infinity, see F. D. Gakhov (1977).
The analog of the characteristic equation on the real axis is the equation of the form
b(x) ∞ x – z 0 ϕ(τ)
a(x)ϕ(x)+ dτ = f(x), (32)
πi τ – z 0 τ – x
–∞
where z 0 is a point that does not belong to the contour. For this equation, all qualitative results
obtained for the characteristic equation with finite contour are still valid together with the formulas.
In particular, the following inversion formulas for the Cauchy type integral hold:
1 ∞ x – z 0 ϕ(τ) 1 ∞ x – z 0 ψ(τ)
ψ(x)= dτ, ϕ(x)= dτ. (33)
πi τ – z 0 τ – x πi τ – z 0 τ – x
–∞ –∞

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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