Oscillatmo- Dirichlet's Integrals. 21 



In considering the integral I/W, introduce tlie relations 



(l-x)-''-' = l + 0(x), 



Then we have 



L' (X) = f'x^. - -i i«s- (V-) [1 + (x)} ""^^-^^-^^ dx 



./ ^ ' ■' X 



^^r, ) /"* a - mi log (l/x) f 1 , /a / \-) COS yJ.iC -, 



= COS / / a:"V ° ^ ' M 1 +0 (a;)] arc 



./ a; 



I „,-., } /'* t -milog(i/.r) f 1 , n/ >f sin /a- , 



+ sin / / a^ e °^ ' ' {[+C)(x)\ ax 



/ a- 



say. Then, by Theorems III and IV, we have, for 0<«<1, 



fx-e-"^' ^"^^ (^/^^ ^^— f?a; r-' r(« + mi) cos [i(« + mi);r] ;.-«ö-n^i log).^ 



/"Ve-"^^^°"(^^''^— ^--^x <-> r(6: + mi)sin [i(« + mi)7r] /-^-milog). 



Hence, we can easily see tliat 



j {?.) <o COS /I /|r/ + mi) COS {|-(a + mi)7r] /"" r -"" log), 



y (?.) ro sin / /"(ft + mi) sin {^(a + mi)~} /"V-"" log'- 



and 



I„'(k) rJ /'(öf + mi)cos {/i — |-(a + miV] /"V-milog).^ 



Since 0<«<1^ evidently we have I/>l2' as x-^x . 

 Thus we obtain 



Z, (;.) = r[a + mi) cos {/ — ^(a + mi)?:} /r^c - mi log lÇl-\. e',) 

 = r(s) cos (/-is7r) /-»(I + e',\ 



where liin ^a' = 0- 



