MR. O. N. WATSON: A THEORY OF ASYMPTOTIC SKItlKS. 311 



Now suppose that we can find a n-.-tl quantity v such that |v<SX 0, 

 R {z exp ( iV)} > y+ 1 ; then we may write 



F (')=f ( , 



" \ 9. 



and, on integrating by parts, we get 



i.e., 



Y(z) = a l ,+ ^ + ... + ^ 



Z - 



where 



|R,.z"| < (n+1)! K |<-zp (ypr 1 sin ff)} {p t cosec 0}" \ exp {- \t\ }d\t\, 



M-r) 



i.e., 



| R.Z" | < (n+ 1 )! K' (p, cosec 0)", when? K' is finite. 



Now 



so that. 



l . (n+l)! + (u-f 2)! K'(p, cosec 0)'}\z\ 1 . 



If p 3 > pi, we have 



p- (n + 1 ) < K V. p," (n + 1 ) (n + 2) < K' >/, 



when; K", K'" are finite and independent of n. 



Therefore 



< B (p t cosec 0)' . n! | z | --'. 



That is to siiy, for values of z such that 



R{zexp(-tV)} 



where \v\ is less than or equal to X 0, F(z) has an asymptotic expansion with 

 grades equal to unity, a radius p, and an outer radius p, cosec 0, where p, is any 

 quantity greater than p. This is, effectively, the result stated at the beginning of 

 the section. 



1 1 . We shall conclude by investigating the characteristics of the asymptotic 

 expansion of a function connected with the "logarithmic integral," or "It" function, 

 defined by the equation 



