Oscillating Dirichlet's Integrals. 



(i) A-) = -(TZ^v^'"""' ' 



A being a certain constant of the form Ä=ae''\ 



My results in Case (i) are as follows: — 



Let a>0 and p be any real constant, then the behaviour of ci^. as 

 w->x , is determined asymptotically as follotvs : 



jy q = ], a = 7r, 



(9) a„ <- J- a-^'^-^e-^" 11^'-^ sin [2ahii-{^p-^)7:]* ; 

 if 0<g<:l, « = (l+g)|-, 



(10) a„^ ^ (gaj' 1+' 71 i+? é-a;^ [Äni+^ -(i^-ïKj^ , 



^[•Z{[+q)7:] L -I 



where ^- = d+'/k '+' « '■"' J 



i/ < (/ < 1 , a = (8 - (/ )-^-, 



V'[2(l + <7);rj L_ -» 



^ Äem^ ^Äe same as that in the above formida. 



Similar results were obtained in Case (ii) and Case (iii). 



Now it may be remarked that, owing to the restricted 

 apphcability of the method, asymptotic formulae were obtained 

 only in the following three cases: 



(1») ,^ = 1, a = 71, 



{'1") 0<q<\, a = (i-]-q^^, 



(S") 0<q<\, a = iS-q)^. 



■ Observe that this becomes Feièr's formula, if we put « = 1. 



