Page 637 - Handbook Of Integral Equations

of the polynomial P ν+m (z) are chosen in accordance with the above conditions, then formulas (51)
give a solution of the nonhomogeneous problem (48) in the class of bounded functions.
Consider another way of constructing a solution, which is more convenient and based on the
construction of a special particular solution.
By the canonical function Y (z) of the nonhomogeneous problem we mean a piecewise analytic
function that satisfies the boundary condition (48), has zero order everywhere in the finite part of
the domain (including the points α k and β j ), and has the least possible order at infinity.
In the construction of the canonical function, we start from the solution given by formulas (51).
Let us construct a polynomial U n (z) that satisfies the following conditions:

(i)
U (β j )= Ψ +(i) (β j ), i =0, 1, ... , p j – 1, j =1, ... , κ,
n
(l)
U (α k )= Ψ –(l) (α k ), l =0, 1, ... , m k – 1, k =1, ... , µ,
n
where Ψ +(i) (β j ) and Ψ –(l) (α k ) are the values of the ith and the lth derivatives at the corresponding
points. Thus, U n (z) is the Hermite interpolation polynomial for the functions

+
Ψ (z) at the points β j ,
Ψ(z)= –
Ψ (z) at the points α k
with interpolation nodes β j and α k of multiplicities p j and m k , respectively (see Subsection 12.3-2).
Such a polynomial is uniquely determined, and its degree is at most n = m + p – 1.
The canonical function of the nonhomogeneous problem can be expressed via the interpolation
polynomial as follows:
–
+
Ψ (z) – U n (z) – – Ψ (z) – U n (z)
+
+
Y (z)= X (z) , Y (z)= X (z) . (52)
κ µ

(z – β j ) p j (z – α k ) m k
j=1 k=1
To construct the general solution of the nonhomogeneous problem (48), we use the fact that
this general solution is the sum of a particular solution of the nonhomogeneous problem and of the
general solution of the homogeneous problem. Applying formulas (47) and (52), we obtain
µ

+
+
+
Φ (z)= Y (z)+ X (z) (z – α k ) m k P ν–p (z),
k=1
κ (53)
–
–
–
Φ (z)= Y (z)+ X (z) (z – β j ) P ν–p (z).
p j
j=1
For the case in which ν – p < 0, we must set P ν–p (z) ≡ 0. Applying formula (52), we readily find
–
–
that the order of Y (z)atinfinity is equal to ν – p +1. If ν < p – 1, then Y (z) has a pole at infinity,
and the canonical function is no longer a solution of the nonhomogeneous problem.
However, on subjecting the constant term H(t)to p – ν – 1 conditions, we can increase the order
of the functions Y (z)atinfinity by p – ν – 1 and thus again make the canonical function Y (z)a
solution of the nonhomogeneous problem. Obviously, to this end it is necessary and sufficient that
in the expansion of the function Ψ(z) – U n (z) in a neighborhood of the point at infinity, the first
p – ν – 1 coefficients be zero. This gives just p – ν – 1 solvability conditions of the problem for
the case under consideration. Let us clarify the character of these conditions. The expansion of
Ψ(z) – U n (z) can be represented in the form
n
Ψ(z) – U n (z)= –a n z – a n–1 z n–1 – ··· – a 0 + a –1 z –1 + a –2 z –2 + ··· + a –k z –k + ··· ,

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 620