Oscillatiny; Diriohlefs Intograls. 75 



The conditions (j<x^o"ln', /><.jc<t', />< ff' give respectively 



a ^ — h (* < 1' « < 2. 

 The equation / + ^'(0) = gives 



" = h- ) ■ 



and we have 



ß = U) + n{6) = 2(w;.)l 



Hence we have: 



If _| < a < 1, then 



IiW "^ i -*?>r i -U^"-^ cos (2w* ;.^ + ^tt), 



!,(;.) .^ i;r^wr i«- i /i'- * sin (2m* ?i + i-) , 

 CC/^O ^ i7r4m-i'*-^-/>-^exp [(2?/i*;.i + i;r)i] ; 



if — -I < a < 2, then 



J.,(/) f-» ^-*m-*"-*>^*''-^'^cos(2m*/*-^;r), 



J,(/) --^ i^;r*w-*''"T/i'-^sin(2w4/* — Itt), 



We observe that all these integrals tend to zero as ;>^x if a < h 

 and they oscillate if a ^ i- 



42- These results may l)e verified as follows. 

 Hardy proved* that 



/"* / 3-\ du 



I cos II COS I -'— -T— r 



./o \ U J It 



=: ^ ^^"^ [ - J\'2/3) + J-''(2y9)-e-4-- J ^(2'i/9) + e*"'^ J -"(2^/9)] . 



4 Sin ^VTT 

 * Messenger of mathematics Vol. XL, ^jjj. Ji et seq. 



