1910-11.] 
On Fourier’s Repeated Integral. 
563 
Section 1. 
On the Fourier Sine-integral J*dvJp sin uv du, when the Limits 
p and q of Integration with respect to n are finite. 
§ 2. Theorem 1. — The repeated integral 
Jo dvfi p f(u) sin uv du , 
where p and q are finite quantities, positive, negative, or zero, necessarily 
■exists, and is equal to 
f’Mdu, 
Jp u 
provided only the latter integral exists. 
Since is summable, so is f(u), and therefore also | f(u) \ . Hence 
the repeated integrals of \f(u)\, taken over a finite rectangle, exist and 
are equal. Also sin uv is bounded. Hence, by a known theorem,* the 
repeated integrals of | f(u) \ sin uv, and therefore of f(u) sin uv, exist and 
are equal over the same finite rectangle. We may therefore write 
( dv f f(u ) sin uv du = f du( 
J 0 J p J p Jo 
f(u) sin uvdv = [ 
Jp u 
cos Bu)du. 
l f(u) 
cos B u du has, as B 
rq 
But, by a theorem of Riemann-Lebesgue,f I 1 
J p u 
increases, the unique limit zero, whence the required result at once follows. 
§ 3. If, as throughout the present paper, f(u) is assumed a priori to be 
summable in every finite interval, it is clear that the only doubt as to the 
existence of the integral in the theorem of the preceding article, relates 
to the behaviour of the function f(u) in any interval, however small, 
containing the origin. But, further, it is not necessary for the existence 
of the repeated integral that the function f(u)/u should be summable ; it 
is sufficient that (f(u)—f( — u))[u should be summable. This is of some 
importance if we reflect that, otherwise, the addition of a constant to f(u) 
would invalidate the existence of the repeated integral. The corresponding 
theorem is stated most conveniently when the limits of integration with 
respect to u differ only in sign ; but evidently the theorem may be suitably 
modified so as to embrace any case. 
* W. H. Young, “On Change of Order of Integration in a Repeated Integral,” 1909, 
Carnb. Phil. Trans., vol. xxi. p. 364. 
t B. Riemann, “Ueber die Darstellbarkeit einer Funktion durch eine tngonometrische 
Reihe,” § 10 (1854), Ges. Werke , 2nd ed., p. 254 ; H. Lebesgue, Legons sur les series 
trigonometriques (1905) ; see also Hobson’s Theory of Functions of a Real Variable , p. 674. 
