which is true, even when the quantity under the integral sign becomes infinite for 
particular values of x, provided the integral be replaced by its principal value, that is, 
provided it be considered as the limit of 
f + f ) ifrx [X U r y V (x + ra)] dx, 
\Ja+e Jo J 
J yfrx [2 U r y F (x + ra)] dx; 
where a, or one of the limiting values a, 0, is the value of x, for which the quantity 
under the integral sign becomes infinite, and e is ultimately evanescent. 
In particular, taking for simplicity a = tt, suppose 
■yfr (x + 7r) = + y\rx, or yfr (x + rir) = (+) r -\Jrx ; 
then observing the equation 
v(+) r 1 _ 
= cot x, or = cosec x, 
x + rir 
according as the upper or under sign is taken, and assuming 'kx — x~ tl , we have finally 
yjrxdx (—)^ _1 
/ 
— 00 X* r/i J, 
y\rxdx _ (—)' A ~ 1 Y 
yjrx 
d Y~' 
cot x 
dx, 
of- 
dxj 
cosec x 
dx, 
the former equation corresponding to the case of yjr (x + tt) = yjrx, the latter to that of 
yfr (x + 7r)= — \jrx. 
Suppose yjr / x = -\Jrgx, g being a positive integer. Then 
y~ t xdx 
x** 
= r 1 1 
J — c 
yjrxdx 
x* 
also if \jr(x + 7t) = yjrx, then yj/ / (x + tt) = yjr,x ; but if yjr (x + 7r) = — y\rx, then -yJ/^x+Tr) 
= ± 'l r / x > the upper or under sign according as g is even or odd. Combining these 
equations, we have 
■yfr(x + tt) = yfrx, g even or odd, 
f *±gxdx_(-y- 1 [* 
J -» of r (g) 
yfr (x + 7t) = — yjrx, g even, 
g* 1 yfrx 
d Y~' 
adx) 
cot X 
' cot 
dx ; 
r 
yjrgxdx _ (-y- 1 [" 
of- Tg 
yfr (x + 7r) = — yjrx, g odd, 
J 0 
dY- 1 
\dxj 
cosec x 
(—Y~ 1 C* 
dx= ~iyr), *> x 
d Y~ 
~dxj 
cosec x 
dx 
yfrgxdx (—)'*~ 1 
f yfrgxdx _ 
J-» 
r g 
g* 1 I yfrx 
dY~ l 
\dx) 
cosec x 
cosec x 
dx. 
the numb 
The 
integrals, 
In pi 
also "T (x, 
where 
2 extend: 
notation 
Functions 
of the an