Page 617 - Handbook Of Integral Equations

we arrive at the notion of the Cauchy principal value of a singular integral.
The Cauchy principal value of the singular integral

dx
b
, a < c < b
x – c
a
is the number
c–ε dx b dx
lim + .
ε→0 x – c x – c
a c+ε
With regard to formula (5), we have

dx b – c
b
=ln . (7)
x – c c – a
a
Consider the more general integral
ϕ(x)
b
dx, (8)
x – c
a
where ϕ(x) ∈ [a, b] is a function satisfying the H¨ older condition. Let us understand this integral in
the sense of the Cauchy principal value, which we define as follows:

ϕ(x) ϕ(x) ϕ(x)
b c–ε b
dx = lim dx + dx .
x – c ε→0 x – c x – c
a a c+ε
We have the identity

b b b
ϕ(x) ϕ(x) – ϕ(c) dx
dx = dx + ϕ(c) ;
x – c x – c x – c
a a a
moreover, the first integral on the right-hand side is convergent as an improper integral, because it
follows from the H¨ older condition that

A
ϕ(x) – ϕ(c)
< , 0 < λ ≤ 1,
x – c
|x – c| 1–λ
and the second integral coincides with (7).
Thus, we see that the singular integral (8), where ϕ(x) satisfies the H¨ older condition, exists in
the sense of the Cauchy principal value and is equal to

ϕ(x) ϕ(x) – ϕ(c) b – c
b b
dx = dx + ϕ(c)ln .
x – c x – c c – a
a a

Some authors denote singular integrals by special symbols like v.p. (valeur principale). How-
ever, this is not necessary because, on one hand, if an integral of the form (8) exists as a proper or an
improper integral, then it exists in the sense of the Cauchy principal value, and their values coincide;
on the other hand, we shall always understand a singular integral in the sense of the Cauchy principal
value. For this reason, we denote a singular integral by the usual integral sign.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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