686 Chapter^9
(c') (PV)l::0J(x)dx = limR-->ool~Rf(x)dx
The abbreviation (PV) before the integral is sometimes dropped when it
is clear from the context that the principal value is to be understood. We
remark that if 1: 0 f(x) dx converges in the sense of the definition given in
( c) its value coincides with its principal value as defined in ( c'). That the
converse is false can be seen by considering the example f(x) = x.
In general, for odd functions the principal value of the integral is zero
since
l
o f(x)dx = {O f(-x)(-dx) = -1R f(x)dx
-R JR O
On the other hand, for even functions we have
1
° f(x) dx = {° f(-x)(-dx) = 1R f(x)dx
-R JR 0
so that
(^1) /
2 1R f(x)dx=1° f(x)dx=1Rf(x)dx
-R -R O
and the existence of the principal value implies the convergence of both
l~ 00 f(x)dx and (^10)
00
f(x)dx. Hence in this case l~ 00 f(x)dx converges in
the ordinary sense ( c).
- Improper integrals of the second kind. For functions having an infinite
discontinuity at a, or at b, or at c (a < c < b) but are otherwise Riemann
integrable on every closed interval, they are defined, respectively, as follows:
(a) l: f(x) dx = lime__.o+ l:,_J(x) dx if the limit exists
(b) l: f( x) dx = lime__.o+ 1:-e f ( x) dx if the limit exists
( c) l: f ( x) dx = lime. 0 + 1:-e f ( x) dx + limei . 0 + f:+e' f ( x) dx if both
limits exist (with f' independent of e)
Again, the terms convergent and divergent are applied to these integrals
accordini~ to whether the corresponding limits do or do not exist. Integrals
of the second kind can be transformed into integrals of the first kind by an
appropriate change of variable. For instance, in case (a) above it suffices
to let x = a + (1/t).
If in case ( c) the integral diverges it may happen that the integral con-
verges in the sense of a principal value, as defined in (9.10-7), i.e., by
choosing e^1 = e. Again, the abbreviation (PV) is sometimes dropped when
the meaning is clear from the context. If f ( x) has a finite number of dis-
continuities in [a, b], near which the function is unbounded, then each of
those discontinuities is to be treated separately as above.