304 MR. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



Also, we notice that, by CAUCHY'S theorem, 



1 f i 



. i ""Vcte = the residue of ~"~V at the origin = . 

 ZTTIJL n \ 



By making use of the equations (15) in conjunction with this last result we get in 

 succession on integrating by parts : 





. fc (z/t)] L + -L. 



(16) 



each of the terms in square brackets vanishing at both ends of the contour. 

 We shall now estimate the value of 

 At all points on L we have 



a 



(z/t) , dz for any Vftlue 

 P, 



+ 



nlz/t n\z/t 



*or brevity we put 



On the arc of the circle we have 



and lP|Sexp| y <|, 



where varies from -fr-Q to 



