1910-11.] On Fourier’s Repeated Integral. 581 
which, by the theorem of Riemann-Lebesgue (§ 10), has the unique limit 
zero when B is indefinitely increased. This proves the theorem. 
Cor. 1. — If f(u) = g(u)h(u) where JJ|g(u)|du exists, and h(u) oscillates 
finitely at infinity, the same result holds * 
Cor. 2.j- — If J"|-f(u) du j exists, then, supposing the integral 
Jo dvj™f(u) cos uv du 
to exist, it is certainly equal to f(0), provided only that the origin does 
not belong to the exceptional set of content zero at which f(u) is not the 
differential coefficient of its integral. 
We have, in fact, only to combine the present theorem with that of § 8. 
§ 23. Theorem 14. — If in the interval (q, oo) the function f(u) is 
monotone decreasing, and approaches the unique limit zero at infinity, 
then 
\ 0 dv\lf(u) cos uv du = 0. 
Since \ b a cos uv du — (sin vb — sin va)/v, it follows by the Second 
Theorem of the Mean that, if a < b and v > e > 0, 
Jl/W cos uv d u L 2f(a)/e. 
Hence, denoting for shortness the integrand f(u) cos uv by F(w, v), it 
follows, firstly, that J Q F(u, v)du exists for each positive value of v (since, 
by hypothesis, f(u) has zero as unique limit at infinity) ; secondly, it is, for 
all values of v greater than e, a bounded function of v ; and, thirdly, 
\qf(u) cos uv du converges boundedly to its limit for all such values of v. 
Hence we may integrate, and get 
v)du = \*dvLt J*F(«, v)du= U\*dv\^(u, v)du= Lt fcduf?F(u, v)dv 
A=oo A==o A=oo 
f 00 x sin B u - sin eu , 
= / Hu) du, 
J Q U 
which is numerically L 2tt/(Q + 0), using the Second Theorem of the Mean, 
and noticing that 
f Q ' sin &u^ u _ j P Q/ 
J Q U J kQ 
sm u 
du 
*q u 
Hence, 
Jf dv\^(u, v)du - 2tt/'(Q + 0) L \ B e dv\™¥(u, v)du L Jf dy F (u, v)du + 2tt/(Q + 0). 
* This includes that case of Pringsheim’s theorem, loc. cit., p. 399, § 5, in which the first 
of Pringsheim’s conditions at infinity (p. 391) is satisfied by one of the factors of the 
integrand. The other factor is, in consequence of Pringsheim’s further assumptions, a 
hounded function of all the variables. 
+ This may be compared with a theorem given by Michel Plancherel on p. 42 of his 
“ Contribution a l’etude de la representation d’une fonction arbitraire par des integrales 
definies,” 1910, Rend, di Palermo , xxx. pp. 1-47. In Plancherel’s theorem J 0 [f{u)\ 2 du 
exists, instead of, as here, J 0 j f{u) | du. 
