Oscillât) n<j,- Dirichlet's Intej^rals. 43 



Now /(^)=[1+^W]^>0. 



since o"{6) > 0, Y/(ä) > 0, J>(d) < 1, a'{d) + / = 0. 



And fie + e) = [ 1 + r(6 + s)] i^'l - <r'(d + e) - ; 



since (t"(0') > 0, ;>'(/9) > 0, £ /7(^) < ^(^). 



Therefore it follows that 6 <:â + t< a. 



25- Integral Ji'\ We now consider the integral 



J-/--') = /' '^ 'iM. cos .// clx = J/ + J"/', 



where f J' = /"^"''^''^'^'Hpixl c osy \ . 



\ ' Jß \ X ' J+(t\x) I ^' , 



' .//Î \ X à + (t\x)} -^ 



ß = Ad + a{d). 



Now let us consider the following difference of integrals 



(42) i' = / ,/'-7 , cos 2/ ^y 



which may be Avritten in the form 



(43) J = ;■_,.;, cos ij dy + ^-^-^ / yXti) cos 7/ dy. 



+ (y\x) 



1 1 



where r(^) = ^, Z(^) - ^^^ - -^^^^.-.^^j^^^. 



