r 
103] 
ON CERTAIN DEFINITE INTEGRALS. 
.'15 
On the subject of the preceding paper may be consulted the following memoirs by 
Raabe, “ Ueber die Summation periodischer Reihen,” Crelle, t. xv. [1836], pp. 355—364, 
and ,£ Ueber die Summation harmonisch periodischer Reihen,” t. ххш. [1842], pp. 105— 
125, and t. xxy. [1843], pp. 160—168. The integrals he considers, are taken between 
the limits 0, oo (instead of — oo, oo). His results are consequently more general than 
those given above, but they might be obtained by an analogous method, instead of 
the much more complicated one adopted by him: thus if ф(х + 2тг) = фх, the integral 
г 00 ¿¡ x 
фх — reduces itself to 
Г , dx Г* Г1 , v œ / 1 1 
фх—-ä— = ахфх\-+2, 1 — | 0 - s 
J о # + 2г7г Jo т [x \æ + 2г7г 2г7 
provided I dx(j)x= 0. The summation in this formula may be effected by means of 
J 0 
the function T and its differential coefficient, and we have 
ralliéL 
J o « 27tJo i X \ 
which is in effect Raabe’s formula (10), Crelle, t. xxv. p. 166. 
By dividing the integral on the right-hand side of the equation into two others 
whose limits are 0, 7r, and ir, respectively, and writing in the second of these 27t — x 
instead of x, then 
( 
. dx 
фх — = 
Г' 
or reducing by 
27г, 
Г' 
фх* 
27Г/ 
Г' 1- 
/ X 
+ Ф (27Г — х) 
г (1 
dx 
ГМ - 
27Г/ 
— 7Г COt \х, 
Г 1 
we have 
, dx , 
фх — = А 
r x г 
Г' 1 - 
фх cot \х dx — — I [фх + ф (27г — х)] ' 
27Г J О 
dx, 
Г 1 - 
which corresponds to Raabe’s formula (10'). If 0 (— x) = — cf)X, so that <jjx + cf) (27r — x) = 0. 
the last formula is simplified ; but then the integral on the first side may be replaced 
r 00 dx 
by j (fix —, so that this belongs to the preceding class of formulae. 
— 00