Page 524 - Handbook Of Integral Equations

If H(u) is a rational function as well, then the jump problem can readily be solved:

Q – (u)
–
+
W (u) – W (u)= H(u). (45)
R – (u)
+
To this end, it suffices to decompose the right-hand side into the sum of partial fractions. Then W (u)
–
and W (u) are the sums of the partial fractions with poles in the lower and the upper half-planes,
respectively. We can directly apply the continuity principle (the analytic continuation theorem) and
the generalized Liouville theorem to the resulting relation
Q – (u) + + R + (u) – –
Y (u) – W (u)= Y (u) – W (u).
R – (u) Q + (u)
The only exceptional point at which the analytic function, which is the same on the entire complex
plane, can have a nonzero order is the point at infinity, at which the order of the function is equal to
ν – 1= m + – n + – 1= n – – m – – 1.
For ν > 0, the solution can be written in the form
R – (z) + – Q + (z) –
+
Y (z)= [W (z)+ P ν–1 (z)], Y (z)= [W (z)+ P ν–1 (z)].
Q – (z) R + (z)
For ν ≤ 0 we must set P ν–1 ≡ 0; moreover, for ν < 0 we must also write out the solvability
conditions that can be obtained by equating with zero the first |ν| terms of the expansion of the
rational function W(z) in a series (in powers of 1/z) in a neighborhood of the point at infinity.
The solution of the jump problem (45) can be obtained either by applying the method of
indeterminate coefficients, as is usually performed in the integration of rational functions, or using
the theory of residuals of analytic functions. Let z k be a pole, of multiplicity m, of the function

Q – (z)/R – (z) H(z). Then the coefficients of the principal part of the decomposition of this function
in a neighborhood of the point z k , which has the form
c k c k
1 m
+ ··· + ,
z – z k (z – z k ) m
can be found by the formula
1 d j–1 Q – (z)
k
c = H(z) .
j j–1
(j – 1)! dz R – (z)
z=z k
The above case is not only of independent interest, as it frequently occurs in practice, but also of
importance as a possible way of solving the problem under general assumptions. The approximation
of arbitrary coefficients of the class under consideration by rational functions is a widespread method
of approximate solution of the Riemann boundary value problem.

10.4-6. Exceptional Cases. The Homogeneous Problem
Assume that the coefficient D(u) of a Riemann boundary value problem has zeros of orders α 1 , ... , α r
at points a 1 , ... , a r , respectively, and poles* of the orders β 1 , ... , β s at points b 1 , ... , b s (α 1 , ... , α r
and β 1 , ... , β s are positive integers). Thus, the coefficient can be represented in the form
r

(u – a i ) α i
r s
i=1
D(u)= D 1 (u), D 1 (u) ≠ 0, –∞ < u < ∞, α i = m, β j = n. (46)
s
i=1 j=1
(u – b j ) β j
j=1
* For the case in which the function D(u) is not analytic, the term “pole” will be used for points at which the function tends to
infinity with integer order.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 506