1910-11.] Sommerf eld’s Form of Fourier s Repeated Integrals. 587 
XLI. — On Sommerfeld’s Form of Fourier’s Repeated Integrals. 
By W. H. Young, Sc.D., F.R.S. Communicated by Professor G. A. 
Gibson. 
(MS. received January 4, 1911. Read February 6, 1911.) 
§ 1. In his treatise on Fourier Series and Integrals Carslaw quotes 
without proof Sommerfeld’s theorem that 
Id f dv f f(u) cos uv e~ hm du — — /( + 0), 
t=oJ o J o 2 
when the limit on the right-hand side exists.* In applied mathematics, 
he remarks, it is this limit, rather than the corresponding Fourier repeated 
integral ^ dv\ p 0 f(u) cos uv du, which occurs. 
In the present paper I propose to extend this result in various ways. 
After proving Sommerfeld’s result on the general hypothesis, not considered 
by him, that the integral is a Lebesgue integral, I show that the limit 
in question is 5-/(0), whenever the origin is a point at which f(u) is the 
differential coefficient of its integral, and I obtain the corresponding results 
for 
l^d^flu) sin uv e~ kvn du. 
In all their generality these statements are only true when the interval 
(0, p) is a finite one. I then show how, under a variety of hypotheses with 
respect to the nature of f(x) at infinity, they can be extended so as to be 
still true when p = + oo . These hypotheses correspond precisely to those 
which have been proved f to be sufficient for the corresponding statements 
as to the Fourier sine and cosine repeated integrals in their usual forms, 
\™dv[™f(u) cos uv du and j™dvj™f(u) sin uv du. 
They are as follows : — 
(i-> I f(u) I du exists, 
* Of. Carslaw, op. cit ., p. 186. The theorem as there stated differs only in form from 
the above, from which it may be easily deduced. The proviso as to the existence of the 
limit /( + 0) is moreover, though tacitly assumed, not expressly stated in the enunciation. 
t A. Pringsheim, “Ueber neue Giiltigkeitsbedingungen fur die Four iersche Integral- 
forinel,” 1909, Math. Ann., lxviii. pp. 367-408. W. H. Young, “On Fourier’s Repeated 
Integrals,” 1910, presented to the Royal Society of Edinburgh. 
