1910-11.] SommerfelcTs Form of Fourier’s Repeated Integrals, 589 
For, since f(u) is a summable function of u alone, it certainly possesses 
finite repeated integrals of equal value, therefore,* since e~ kv<lt cos uv is a 
bounded function of (u, v), 
lo H e ~ lv t f( u ) cos uv du = Jo du^ e~ kvH f(u) cos uvdv . . ( 1 ) 
Again, since 
| | r B e -kv"-t cos uvdv \ £ f*e~ kv ' 2f dv, 
and is therefore a bounded function of (B, u) in the interval 0 L B L oo , 
0 Lu Lp, and f(u) is summable, it follows f that, if in (1) we let B approach 
infinity, we may, on the right, introduce the limit under the first sign of 
integration, since ^e~ kv2t cos uv dv exists. We thus get 
JJc?y[o e~ kv ' 2 f(u) cos uv du = JJ du f{ff)\™e~ kvH cos uv dv 
|lo 
= J^\ P 0 ViR f{uJikt)e- u, du. . . (2) 
Now, JJ e~ u2 du exists and Jir, and f(u J^ht) converges boundedly to 
/( + 0). Hence j the above expression has an unique limit, when the two t’ s 
which occur are regarded as approaching their limit zero independently. 
Thus we get the required equality. 
^ 0R * e~ kv%t f(u) cos uv du = 0, 
'provided p and q have the same sign, and f(u) is summable in the interval 
(p , q). This follows at once from the above theorem, putting f(u) = 0, 
when u<p. 
§ 3. Theorem. — If f(u) is a summable function , and the origin does 
not belong to the exceptional set of content zero at which f(u) is not the 
differential coefficient of its integral, 
Lt I dv f e~ M>t f(u) cos uv du = ~f(0). 
As in the preceding proof we obtain equation (2), in which for shortness 
* W. H. Young, “On the Change of Order of Integration in an Improper Repeated 
Integral,” 1910, Trans. Gamb. Phil Soc ., vol. xxi. p. 364, § 4. 
t By the theorem enunciated at the end of § 1. 
I W. H. Young, “On a Theorem of Tannery’s and its Analogue in the Theory of 
Definite Integrals,” 1910, Mess. Math., p. 58, Cor. 2. See also Bromwich’s Introduction to 
the Theory of Infinite Series (1908), p. 438, where the condition of uniform convergence 
may be changed to bounded convergence. 
