Page 523 - Handbook Of Integral Equations

provided that |ν| conditions (36) hold. Here we have

–ν
1 ∞ u – i –iux
g(x)= √ ln D(u) e du, (40)
2π –∞ u + i
1 ∞ 1 0
+ izx – izx
G (z)= √ g(x)e dx, G (z)= – √ g(x)e dx, (41)
2π 0 2π –∞
z – i –ν
–
+
+
–
X (z)= e G (z) , X (z)= e G (z) , (42)
z + i

1 ∞ H(u) –iux
w(x)= √ + e du, (43)
2π –∞ X (u)
0
1 ∞ 1
+ izx – izx
W (z)= √ w(x)e dx, W (z)= – √ w(x)e dx. (44)
2π 0 2π –∞
The sequence of operations to construct a solution can be described as follows.
◦
1 . By virtue of formula (40) we find g(x), and then, with the help of (41), for the given g(x)we
±
find G (z).
2 . By formulas (42) the canonical function X (z) is determined.
±
◦
◦
±
3 . By formula (43) we determine w(x), and then apply formula (44) to find W (z).
After this, solutions of the homogeneous and nonhomogeneous problems can be found by
formulas (37)–(39) and (42). For the case ν < 0, it is also necessary to verify the solvability
conditions (36).
10.4-5. Problems With Rational Coefficients
The solution of the Riemann problem thus obtained requires evaluation of several Fourier integrals.
This can also be readily expressed by means of integrals of the Cauchy type. As a rule, the integrals
cannot be evaluated in the closed form and are calculated by various approximate methods. This
process is rather cumbersome, and therefore it is of interest to select cases in which the solution can
be obtained directly from the boundary condition by applying the method of analytic continuation
without using the antiderivatives.
Assume that in the boundary condition (14) we have
R + (u) R – (u)
D(u)= .
Q + (u) Q – (u)

Here R + (u) and Q + (u)(R – (u) and Q – (u)) are polynomials whose zeros belong to the upper (lower)
half-plane (we must avoid confusing these polynomials with the one-sided functions introduced
above, which have similar notation). Denote the degrees of the polynomials P + , R – , Q + , and Q –
by m + , m – , n + , and n – , respectively. Since, by the assumption of the problem, the value D(∞) can
be neither zero nor infinity, it follows that the relation m + + m – = n + + n – holds. The index of the
problem can be expressed by the formula

ν = Ind D(u)= m + – n + = –(m – – n – ).

On multiplying the boundary condition by Q – (u)/P – (u) we obtain

Q – (u) + R + (u) – Q – (u)
Y (u) – Y (u)= H(u).
R – (u) Q + (u) R – (u)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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