585 
1910-11.] On Fourier’s Repeated Integral. 
For, in this case, ( ; - J du necessarily exists, provided / does so. 
§ 26. As in § 20, we can extend the preceding theorems to the case 
when g(u) has bounded variation in the whole infinite interval ; and q(u), 
the total variation of g(u) measured from infinity, after division by u, can 
be integrated from q to infinity. We thus have the following theorem : — 
Theorem 18. — If throughout the interval (q, oo ), 
f(u) = g(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 I may be integrated from q to infinity 
f*d/offi(u) cos uv du = 0, 
provided, in addition, when h(u) is not an even function, y(u), that the 
total variation of g(u) measured from infinity, after division by u, may 
be integrated from q to infinity. 
§ 27. — As in discussing the corresponding theorems for the sine-integral, 
it is evidently unnecessary for the argument of the preceding article that 
the square of h(u) should be summable, provided the series whose general 
term is ( | a n | + | b n | }{n converges. Thus we have, as in Section 3, § 18, an 
extended theorem, as follows : — 
Theorem 18, bis. — If for all values of u in the open interval (q, oo ), 
f(u) = g(u)h(u), 
where g(u) has bounded variation in the whole infinite interval, and 
approaches the limit zero at infinity, while h(u) is any summable function 
whose Lebesgue integral, proper or improper, can be expanded in a series 
of the form 
fi'h(u)du = A 0 u + 5 (A n cos k n u + B„ sin k n u), 
the positive quantities k n increasing without limit with n, and the series 
whose general term is j A n + B n | converging, then, if / j — — ! du exists, 
J q I U 
j>J>) sin uv du = 0, 
provided either the coefficients B n are identically zero, or J ~pdu exists, 
y(u) denoting the total variation of g(u) measv^red from infinity. 
§ 28. As at the corresponding point in Section 3 (§ 19), we have now 
the following corollary : — 
