Page 665 - Handbook Of Integral Equations

Relation (12) becomes
+
–
+
+
+
–
Π (t)ϕ (t)+ A [ϕ (t)] = Π (t)[D(t)ϕ (t)+ H(t)], (14)
+
+
+
–
where A is the integral operator with kernel A (t, τ). Since the function A [ϕ (t)] is analytic
+
on Ω , it follows that the last relation is an ordinary Riemann problem for which the functions
+
+
–
–
+
Π (t)ϕ (t)+A [ϕ (t)] and ϕ (t) can be defined in a closed form, and hence the same holds for ϕ(t).
+
–
±
Namely, let us rewrite the function D(t) in the form D(t)= X (t)/X (t), where X (z) is the
canonical function of the Riemann problem, and reduce relation (14) to the form in which the
generalized Liouville theorem can be applied (see Subsection 12.3-1). We arrive at a polynomial
of degree at most ν – 1+ n m k with arbitrary coefficients (for the case in which ν + n m k > 0).
k=1 k=1
+
+
+
However, the presence of the factor Π (t) (on ϕ (t)), which vanishes in Ω with total order of zeros
m k , clearly reduces the number of arbitrary constants in the general solution.
n
k=1
Remark 1. Following the lines of the discussion in Subsection 13.3-2 we can treat the case in
which the kernel K(t, τ) is meromorphic with respect to τ as well. In this case, Eq. (1) can be
reduced to a Riemann problem of the type (12) and a linear algebraic system.
Remark 2. The solutions of a complete singular integral equation that are constructed in Sec-
tion 13.3 can be applied for the case in which the contour L is a collection of finitely many disjoint
smooth closed contours.
Example 1. Consider the equation
1 cos(τ – t)
λϕ(t)+ ϕ(τ) dτ = f(t), (15)
πi L τ – t
where L is an arbitrary closed contour.
Note that the function M(t, τ) = cos(τ – t) has the property M(t, t) ≡ 1. Therefore, it remains to apply formula (6), and
thus for (15) we have
1 1 cos(τ – t)
ϕ(t)= λf(t) – f(τ) dτ , λ ≠ ±1.
2
λ – 1 πi L τ – t
Example 2. Consider the equation
1 sin(τ – t)
λϕ(t)+ ϕ(τ) dτ = f(t), (16)
πi L (τ – t) 2
where L is an arbitrary closed contour.
The function M(t, τ) = sin(τ – t)/(τ – t) has the property M(t, t) ≡ 1. Therefore, applying formula (6), for (16) we
obtain

1 1 sin(τ – t)
ϕ(t)= λf(t) – f(τ) dτ , λ ≠ ±1.
2
λ – 1 πi L (τ – t) 2
•
Reference for Section 13.3: F. D. Gakhov (1977).
13.4. The Regularization Method for Complete Singular
Integral Equations
13.4-1. Certain Properties of Singular Operators
Let K 1 and K 2 be singular operators,
1 M 1 (t, τ)
K 1 [ϕ(t)] ≡ a 1 (t)ϕ(t)+ ϕ(τ) dτ, (1)
πi L τ – t
1 M 2 (t, τ)
K 2 [ω(t)] ≡ a 2 (t)ω(t)+ ω(τ) dτ. (2)
πi L τ – t

The operator K = K 2 K 1 defined by the formula K[ϕ(t)] = K 2 K 1 [ϕ(t)] is called the composition
or the product of the operators K 1 and K 2 .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

Page 648