Mr. Boots on the Analysis of Discontinuous Functions. 131 
—kr 
which is to be regarded as the limit of the more general integral Vi dae 
cos(qxv)a"~', k being positive, the value of the former being, on this assumption, 
) dxcos(qx)a" = oO cos (n=). (1) 
0 2 
We are hence naturally led to the bine: 548 of such integrals as constitute 
the limiting cases of more general forms, involving two vanishing quantities, / 
and k’, Of this kind would be the integral above mentioned, when 7 should be 
negative, as may be shewn by the following analysis. We have 
Bea) 
) due cos(av)v" | = 
J (pa 
2 5 
wena =) (2) 
cos ( mtan—! = p3 
‘ie. ( a5) = a dve—™ cos(av yo", 
(2+ 2°)? T(2) A 
and 
» dx e** cosgx cos( mtan = 
\ f ( am = " dadve*—** cos (qa) cos (vx) vu" 
° (R-+La’)? — T(n) vo 
T ie 
= OT (n) (u+v); (3) 
provided that 
= al \ dvdxe** cos(va—qu) ev", 
THowo 
1c" : 
== \ dvdxe** cos(va-qu)ye vu", 
FTse0LSE0 
expressions of which the limiting values, with reference to k’, are, by Fourier’s 
theorem, 
sec 1g, 0n.0;a8 9-0, OF G.<), 
v = ¢4(—q)"" or 0, as —q > 0, or —¢g < 0, 
or as g < 0, or g >.0, 
“uftvae “gq, or &1(— q)"' as g > 0, org < 0. 
Hence, taking the limit with reference to &, and substituting in (3). Limit of 
, dxe~* cos(qx) cos (n tan“! a) 
0 (e+ 2°)? = 550) 
s 2 
qs © saa (— ae ao he 
