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 :