64 
IMPROPER MULTIPLE INTEGRALS 
f v* T 
To provide for the case that <£ may not be defined for certain 
points of 53 we give the symbol 2) the following definition. 
where T = (5 when the integral 3) is convergent, or in the con 
trary case T is such a part of (5 that 
— «<./’/</3, (6 
and such that the integral in 6) is numerically as large as 6) will 
permit. 
Sometimes it is convenient to denote T more specifically by r a|8 . 
The points 53 aJ 3 are the points of 53 at which 6) holds. It will 
be noticed that each 53 a p in 5) contains all the points of 53 where 
the integral 3) is not convergent. Thus 
53=£7i$af J }. 
Hence when 53 is complete or metric, 
lim 53ais=53- (7 
a, /3=co 
Before going farther it will aid the reader to consider a few 
examples. 
71. Example 1. Let 21 be as in the example in 70, 2, while/ = n 2 
at x — 
m 
We see that 
f /=0. 
(1 
On the other hand 53 a( s contains but a finite number of points 
for any a, /3. Thus 
W=°- < 2 
Thus the two integrals 1), 2) exist and are equal. 
Example 2. The fact that the integrals in Ex. 1 vanish may 
lead the reader to depreciate the value of an example of this kind. 
This would be unfortunate, as it is easy to modify the function so 
that these integrals do not vanish.