Oscillating- Diriohlet's Integrals. g] 



//• I{n) < — , then », = (1/n) ; 



if 1(71) > — , the?i a„ --^ I (71). 



Proof. By Lemma 9, we have 



= I{n) + r{7i) 

 say. Then 



I\n) = J- /"''" /(e«^) e-^'^dd 



= -_L /' '"\u+iV) (cos 7id-i sin nd) dd, 



where f{e'^) = C7+iF, 



fZand F denoting real functions of 6. Since /(^) is regular on the 

 circle of convergence, except at z = }, the functions 



U, V, ^^^ ^^ 



dd ' (Z/? 



have no singularities and are integrable in the interval (ç, 2.rr—$). 

 Hence 



/ (J COS 710 do ^=\ — — / -^^^ sm 710 do 



J s \ 71 \s 71 J ? do 



= 0(l/w). 

 Similarly the integrals 



27:-? r-^-^ 



TJ^innddd, / V iid dd 



