SECT. I.] CERTAIN DEFINITE INTEGRALS. 349 



is equivalent to -^ldusmuu r , the integral being taken from u 

 nothing to u infinite. 

 Let fjL be the integral 



00 



du sin u u r ; 

 o 



it remains to form the integral 



L * rfj 



I a q sin qx -^ LL, or LLX \ du sin u i 



J q J 



denoting the last integral by v, taken from u nothing to u infinite, 

 we have as the result of two successive integrations the term 

 x r fjiv. We must then have, according to the condition expressed 

 by the equation (e), 



| 7T Of = fJLV X f Or JJLV 7T J 



thus the product of the two transcendants 



/*, r . , [ x du ._ . . . 



I aww smw and I u sm w is ^TT. 

 Jo Jo u 



For example, if r = - ^ , we find the known result 



in the same manner we find 



[ducosu 



I -7^- = 2 



Jo ^/u V 2 



and from these two equations we might also conclude the following 1 , 



f 1 - 



I dqe~ q = g -S/TT, which has been employed for some time. 



361. By means of the equations (e) and (e) we may solve the 

 following problem, which belongs also to partial differential 

 analysis. What function Q of the variable q must be placed under 



1 The way is simply to use the expressions e~ = +cos ^-12+ */ -1 sin^/- 1 2, 



transforming a and 6 by writing y* for t&amp;lt; and recollecting that \ - 



Cf. 407. [R. I . E.] 



