62 
NOTE ON THE INTEGRALS J COS X 2 dx AND | sin X 2 dx, 
. 
[568 
where the first term is to be taken between the limits oo, 0; viz. this is 
du 
db 
~.yh 
(—a+bi)y 
o 2 (— a + hi) 
iz u. 
— ci + bi 
When a is not =0, the first term vanishes at each limit, and we have 
du _ — i 
db ~ 2 (— a + bi) U ' 
The doubt was in effect whether this last equation holds good for the limiting value 
a = 0. When a. is =0, then in the original equation for ^ the first term is indeterm- 
du 
inate, and if the equation were true, it would follow that ^ was indeterminate; the 
du . 
original equation for -^r is not true, but we truly have 
du _ 1 
db 2b U> 
the same result as would be obtained from the general equation, rejecting the first 
term and writing a = 0. 
To explain this observe that for a = 0, we have 
u = I y~- e iby dy, 
J o 
which for a moment I write 
u = [ y~^ e iby dy, 
Jo 
where, as before, b is taken to be positive. Writing herein by = x, we have 
1 f bk 
= ml ^ eadx ’ 
M 
and assuming only that the integral I x~^ e ix dx has a determinate limit as M becomes 
J o 
indefinitely large*, then supposing that k is indefinitely large, the integral in the last- 
mentioned expression for u has the value in question 
x~^e ix dx], 
(-/; 
* This is in fact the theorem f e ix2 dx= a determinate value {= £ J(tt) e^ 71 }, proved in the former part 
J 0 
of the present Note.