Page 520 - Handbook Of Integral Equations

Such a representation of D(u) in the form of the ratio of boundary values of the canonical function
is often called a factorization.
Now we consider the homogeneous Riemann problem with the boundary condition

+
–
Y (u)= D(u)Y (u), D(∞) = 1. (25)
On substituting the expression (24) for D(u) into (25) we reduce the boundary condition to the form

–
+
Y (u) Y (u)
= . (26)
+
X (u) X (u)
–
According to formulas (23) for X(z), the left- and the right-hand sides of Eq. (26) contain the
boundary values of functions that are analytic on the upper and lower half-planes, respectively,
possibly except for the point –i at which the order is equal to ν. In the chosen function class, each
function vanishes at infinity. In this case, it follows from the analytic continuation theorem and the
generalized Liouville theorem (see Subsection 10.4-3) that for ν > 0 we have
+
–
Y (z) Y (z) P ν–1 (z)
= = , (27)
+
–
X (z) X (z) (z + i) ν
where P ν–1 (z) is an arbitrary polynomial of degree ν – 1 (the degree of the numerator is less than
that of the denominator because Y(∞) = 0). Hence,
P ν–1 (z)
Y(z)= X(z) . (28)
(z + i) ν

For ν ≤ 0, it follows from Y(∞) = 0 that Y(z) ≡ 0 by the generalized Liouville theorem.
Hence, for ν > 0, the homogeneous Riemann boundary value problem has precisely ν linearly
independent solutions of the form

z k–1 X(z)
, k =1, 2, ... , ν,
(z + i) ν
and for ν ≤ 0, there are no nontrivial solutions.
The right-hand side of Eq. (28) has exactly ν zeros on the entire plane, including the zero at
infinity. These zeros can lie at arbitrary points of the upper and lower half-plane or on the real axis.
Denote the number of zeros on the real axis by N 0 . In the general case (without the requirement that
there are no zeros on the real axis), formula (19) is replaced by the relation

N + + N – + N 0 = Ind D(u)= ν. (29)

Let us pass to the solution of the nonhomogeneous Riemann problem with the boundary condi-
tion (14). We apply relation (24) and reduce the boundary condition to the form

+
–
Y (u) Y (u) H(u)
= + . (30)
+
–
+
X (u) X (u) X (u)
Let us express the last summand as the difference of the boundary values of functions that are
analytic in the upper and the lower half-plane (see the jump problem), that is,
H(u)
+
–
W (u) – W (u)= , (31)
+
X (u)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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