Page 655 - Handbook Of Integral Equations

According to the Sokhotski–Plemelj formulas (see Subsection 12.2-5), we have

–
+
ϕ(t)= Φ (t) – Φ (t),
1 ϕ(τ) + – (3)
dτ = Φ (t)+ Φ (t).
πi L τ – z
+
On substituting (3) into (1) and solving the resultant equation for Φ (t), we see that the piecewise
analytic function Φ(z) must be a solution of the Riemann boundary value problem

–
+
Φ (t)= D(t)Φ (t)+ H(t), (4)
where
a(t) – b(t) f(t)
D(t)= , H(t)= . (5)
a(t)+ b(t) a(t)+ b(t)
Since the function Φ(z) is represented by a Cauchy type integral, it follows that this function must
satisfy the additional condition
–
Φ (∞)=0. (6)
The index ν of the coefficient D(t) of the Riemann problem (4) is called the index of the integral
equation (1). On solving the boundary value problem (4), we find the solution of Eq. (1) by the first
formula in (3).
Thus, the integral equation (1) is reduced to the Riemann boundary value problem (4). To
establish the equivalence of the equation to the boundary value problem we note that, conversely,
the function ϕ(t) that is found by the above-mentioned method from the solution of the boundary
value problem necessarily satisfies Eq. (1).
We first consider the following normal (nonexceptional) case in which the coefficient D(t)of
the Riemann problem (4) admits no zero or infinite values, which amounts to the condition

a(t) ± b(t) ≠ 0 (7)
for Eq. (1). To simplify the subsequent formulas, we assume that the coefficients of Eq. (1) satisfy
the condition
2
2
a (t) – b (t)=1. (8)

2
2
This can always be achieved by dividing the equation by a (t) – b (t).
Let us write out the solution of the Riemann boundary value problem (4) under the assumption
ν ≥ 0 and then use the Sokhotski–Plemelj formulas to find the limit values of the corresponding
functions (see Subsections 12.2-5, 12.3-6, and 12.3-10):

1 H(t) 1 – – 1 H(t) 1
+
+
Φ (t)= X (t) +Ψ(t)– P ν–1 (t) , Φ (t)= X (t) – +Ψ(t)– P ν–1 (t) , (9)
+
+
2 X (t) 2 2 X (t) 2
where
1 H(τ) dτ
Ψ(t)= . (10)
2πi X (τ) τ – t
+
L
1
The arbitrary polynomial is taken in the form – P ν–1 (t), which is convenient for the subsequent
2
notation.
Hence, by formula (3) we have
– –
1 X (t) + X (t) 1
ϕ(t)= 1+ H(t)+ X (t) 1 – Ψ(t) – P ν–1 (t) .
+
+
2 X (t) X (t) 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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