1910-11.] 
On Fourier’s Repeated Integral. 
559 
XL. — On Fourier’s Repeated Integral. By W. H. Young, Sc.D., F.RS. 
Communicated by Professor G. A. Gibson. 
(MS. received December 28, 1910. Read February 6, 1911.) 
Introduction. 
§ 1. Pringsheim has recently reopened the question as to the circumstances 
under which Fourier’s repeated integral exists and represents the function 
to which it corresponds.* In its simplest form the theorem concerning this 
integral asserts that, with provisos to be specified — 
/•oo r oo 
I dv I f{u) cos uv du = — /( + 0), 
Jo jo' '3' 
/( -f 0) being the unique limit, supposed to exist, of f(u) as u approaches 
the value zero. From this equation, indeed, the remainder of the theory 
follows immediately. If it is to be true, certain conditions must be satisfied 
at the origin, at infinity, and in the finite part of the range of values of the 
independent variable. Till Pringsheim’s paper appeared, the only condition 
known to be sufficient at infinity was of a very special character, requiring 
no less than the absolute integrability of the function at infinity. Thus 
the theorem did not, for instance, apply to the function (1 -\-u)~ h . The 
step taken by Pringsheim is an important one. He has shown that it is 
sufficient if the function is a monotone decreasing one, with zero as unique 
limit at infinity. What is even more noteworthy, it appears that it is 
sufficient for the function to be expressible as the product of two factors, 
one, cj), having the character above described, while the other is a cosine, or a 
sine, or, more generally, is expressible in the form of a trigonometrical 
series whose coefficients form an absolutely convergent series, provided 
that in the last two cases the former of the two functions when divided 
by the independent variable is absolutely integrable in the whole infinite 
interval. 
Pringsheim requires certain other conditions to be satisfied by the 
function expressed by the trigonometrical series ; these appear, however, 
only to have been introduced to secure that the product of the two functions 
also satisfies sufficient conditions near the origin. It is more convenient to 
consider the behaviour of the function in the neighbourhood of the origin, 
* A Pringsheim, “ Ueber neue Giiltigkeitsbedingungen fur die Fouriersche Integral- 
formel,” (1909), Math. Ann., lxviii. pp. 367-408. 
