90 



Art. 4. — M. Kuniyeda : 



if q = h 



( çr(^)--^6-««'\n+0(^^-)], 



I </>{()) = hp'+ 0(6^). 



Hence we have 



Avli ere 



/(n) = A eèi'^' / ' W{x) e- {'"■'■ + ''(■'■)]'> rix, 



io(x) = ax"' cos q7T, 

 rjSi,x) = x-Pe-^''-''''''"^''{l + t^x) + is^x)}, 



ej and e- ])eing real, continuoas and differential )le function^^ of x 

 in tlie interval (0, ç), such that 



lim e^ix) = 0, lim e,.-,(_x) = 0. 

 .r^-O .!;->o ' 



Since sinqz>0, uj(x) tends ex]:>onentially to zero as x^O. There- 

 fore the integral J{n) is ahsolutely convergent. Now 



(75) / ü3(x)e~ ^" '■ + '^(•'')^ ' clx = / ÜJ (x) cos a(x) cos nx dx 



,/ ./ 



- / Tjj(x)^\no(x)^\y\iix dx 

 — */ vs {x) co^ a{x) '&\n nx dx 



I 



— ■i / w(«) sill <T(^a;) COS naj <^a;. 



First we coiisider tlic integral 

 / w (r/) cos a{x) cos nx dx 



^ ^(e)cos<l)smnc _ 1 /"^^f ^(^) cos a{x)] sin nrr ^;r, 

 n n / dx 



and 



/ , - [ z5 fic) cos aix) \ sin 7ix dx \ ^ / [\ v3'{x) \ + i w{x)a'{x) \ ] dx. 

 J dx ~ ' ' ./ 



