MR. 0. N. WATSON A THEORY OF ASYMPTOTIC SERIES. 303 



Let us suppose that the expansion is valid when I j f |i y and a ^ arg ,r -^ ft, where 

 /J-a = 7T+2X and < X < TT. 



Putting : = .rexp { (a + )t} and /(.r) = F(z), we see that F(z) possesses an 

 asymptotic expansion of the form* 



where 



|a.|sA.n! 



when |argz|s ^TT + X and |z]iy. 

 We notice that 



where J, = B . n! er"y~'. 



Let tin- region in which tin- asymptotic expansion of F(z)is valid he called the 

 region D. 



Let L be a contour formed hy the following lines : 



(i) The portion of the ray arg z = (far+0) for which |z|iy|fl, being an 

 arbitrary quantity, as small as we please, such that < 6 < X. 



(ii) The major arc of the circle |z| = y|<| terminated by the points 



(iii) The portion of the ray arg z = TT + 6 for which | ; | S y j | . 

 us consider the function tj>(t) defined by the equation 



when; |arg<| S X 0. 



We observe that when z is at any point on tin- contour L, z/t lies within or on the 

 Ixmndary of the region D. 



Now let us define functions /> \jt 3 , ..., fa ..... by the system of equations 



The path of integration is supposed to be taken along the ray from v to infinity 

 which, when produced backwards, passes through the point w = 0; we deduce by 

 continued integration that the asymptotic expansion of \(> n (r) is 



,/, M = ' .ti xtm 



n\v (n-U)!r (n f MI): ,~ 



where 



provided that v lies within the region D or on its boundary. 



* This function F will not be confused with the F of Section 7. 



