568 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
Fourier series of g(u), at the point u — 0. Hence, since the limit in question 
is supposed to exist, the Fourier series converges at the point u = 0. But, 
as we have seen, g(u) is, at the point u = 0, the differential coefficient of its 
indefinite integral. From these two facts we have, by a known theorem,* 
the result that the Fourier series of g(u) converges at the point u = 0 to 
g( 0). Hence 
I = 27n/(0) = 
Con. — If, except for a set of values of x of content zero in the interval 
(-p>p)> . 
Jo dv\ P - P f( u ) cos v ( u ~ x )du 
exists, it is throughout that interval equal to 7 rf(x), except for a set of values 
of x of content zero. 
If x be an internal point of the interval ( — p, p) at which the repeated 
integral exists, the latter is equal to 
+ x) cos uv du = J 0 dv \ P _ p f(u + x) cos uv du. 
Hence, since f(x) is necessarily the differential coefficient of its integral 
except for a set of values of x of content zero,! the required result follows. 
Section 3. 
On the Fourier Sine-integral J 0 dvjj^ f(u) sin uv du. 
§ 9. Here the limits of integration with respect to u are both infinite. 
Hence, if we adopt the usual interpretation, we assume the existence of 
the repeated integral when these infinite limits are replaced by finite 
limits. We accordingly assume further that one of the conditions in 
Section 2 is fulfilled. 
Breaking the repeated integral, then, up into three parts, in one of 
which the limits of integration with respect to u are of opposite signs, 
we reduce the discussion to that of \odv\™f(u) sin uv du, where q is positive 
and not zero, and a precisely similar integral in which the limits of 
integration with respect to u are the same in magnitude, but of opposite 
sign. 
§ 10. In the sections that follow, we shall require the theorem of 
Riemann-Lebesgue , quoted in § 2, in the extended form in which the 
* P. Fatou, “ Series trigonometriques et Series de Taylor,” 1905, Acta Math., xxx. pp. 
335-400; also H. Lebesgue, “Sur les integrates singulieres,” 1910, Ann. de Toidouse , 
p. 90. 
t For this known result see, for instance, my paper on “ Functions of Bounded Variation,” 
1910, Quart. Journ., p. 82. 
