Oscillatiuo- Di ri chiefs Integrals. 91 



But \ v3'ix)\ + \ ôs(x)(t'(^)\ tends exponentially to zero as x->0. 

 Hence the last integral is convergent. Thei-efore we have 



/ üj{x) cos (t(x) cos nx clx = 0(1/;?,). 



./ 



Similarly the other three integrals on the right-hand side of (~~)) 

 assume values of the same type. 

 Thus we obtain 



(76) Jin) = 0(l/w). 



We observe that, in this case, p may tak« any value, positive or 

 negative. 



52- (iii) The case in which 0<:q<'[, « = (3 — g)-^-, p<l + q. 

 In this case 



Hence, by ("()'), 



^'^^) = -J, (1 +ö(ö^-'0], m =- % + hp^+oiß), 



and, if we write 



o{x) = xr^^-^K rf.x) = ax-\ 



then 



Jin) = -l-e^P"" I ^^ [l+ei(.3^) + 'ie„(a-)}e-^'''+^(")^"^a; 



'i/T J X 



9 



where 



\ e^P""' {Jy,(ii)- iJln)}, 



J^{n) = f'-^^{l + h(x) + ie,{x')] cos {nx + (T(x)] doc, 



./ X 



JJn) = / i^(l + ei'a?) + i6.,(a,-)] sin [nx + (T(x)] dx, 



Jo X 



^i{x) and 6o(a; being real functions such that 



t,(x) = 0{x'-'), ^.ix) = 0{x'-'). 



