54
IMPROPER MULTIPLE INTEGRALS
Let us establish the theorem for the upper integral; similar
reasoning may be used for the lower. Since 1) is convergent,
S 9
and
A = lim I h
a, £= x ^S>I a/3
(3
(4
exist by 44, l. Since 3) exists, we have by 53,
o <fg< e v
(5
for any 53 < 21 such that 53 < some <t' .
Since 4) exists, there exists a pair of values a, b such that
= f h + T) , 0< rj < j,
— 2l<z6 4
since the integral on the right side of 4) is a monotone increasing
function of a, b.
Since 21 = 53 + (5 is an unmixed division of 21,
C h= C h+ C h.
~ ^®o/3
lai3
Since h> 0, and the limit 4) exists, the above shows that
/x= lim | h , v= lim j h
a, /3=00 J$} a a a, 0=co ^L<S a o
exist and that
\ — g T" i>.
Then a, b being the same as in 6),
a
= f h + V ',
s'®,-
and we show that
0 < 7)' < 7]
as in 52. Let now c > a, b ; then
(8
(9
f
h< c53<7
“ 4
(10
if we take
53 < -j— = o’".
4 <?
Thus,
But
by 44, l.
57. #
2i such tin
For if we
and (5„ =
Passing t<
58. 1.
If 1°, t
L=«/,
and if 2°,
If 1° ho
gent, then
Let us ]
Z) a , p be a (
£)*/3 lying :