1910-11.] 
On Fourier’s Kepeated Integral. 
579 
g(u) being a monotone decreasing function with zero as limit at infinity , 
provided I clu exists ; or provided the coefficients a n are identically 
J q U 
7 r g(u)h(u) | j . 
zero, and / ^ 7 7 ! clu exists. 
J q U 
§ 20. If g(u) is a function of bounded variation in the whole infinite 
interval (g, oo ) with zero as unique limit at infinity, and P(u) and — N(u) 
are the positive and negative variations of g(u), 
g(u) = P (u) - N(«i) = (W - N(u) ) - ( W - P (u) ), 
where W is the common limit of P(yt) and N(u) when u increases in- 
definitely. W — N(u) and W — P(u) are then both monotone decreasing 
functions of u, with zero as unique limit at infinity, and if either of them, 
when divided by u, may be integrated from q to infinity, so can the other, 
always supposing that g(u)/u may be integrated from q to infinity.* 
The total variation of g(u) is P( / ^) + N(u), which has at infinity the 
limit 2W ; thus 2 W — P(V) — N(V), or, say, y(u), may be called the total 
variation of g(u) measured from infinity. Under the above circumstances, 
therefore, this function is a monotone decreasing function with zero as 
limit at infinity, and, after division by u, may be integrated from q to oo . 
Conversely, if this function y(u), when divided by u, may be integrated 
from q to infinity, so may g(u), as well as its positive and negative 
variations measured from infinity. 
In this way we have at once the following extension of the preceding 
theorems : — 
Theorem 12. — If throughout the interval (q, oo ), 
f(u) = y(u)h(u), 
where g(u) is a function of bounded variation in the whole infinite 
interval, with zero as limit at infinity, and h(u) is any periodic summable 
function whose Fourier coefficients a n and b n are such that the series whose 
general terms are respectively | a n | /n and | b n | /n both converge, then, pro- 
vided | f(u)/u | may be integrated from q to infinity, 
f dvi flu ) sin uv du — f -t^du, 
J o Jq J a u 
* There is a certain want of clearness on this point in Pringsheim’s remarks on p. 397, 
as to his condition b x , referred to again on p. 400 ; if g{u ) is only of bounded variation, 
r 00 yy _ p/ w \ 7“ W - N(V) 
not monotone, the existence of the integrals —du and I — du entails that 
J q. U J q U 
J 
of 
r g (u) 
du , hut it is not a consequence of it. 
