Mi;. <;. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 291 



Using the expansion (6) and keeping the notation of that equation, we find that 



*{/(*)} -'.+ ! + ^ + ... + !+T., (8) 



\\licie <, c,, ..., T. are now defined by the equations 



<-o = </, T = i gjtf, 



M-l 



and, when n > 0, 



c.= i7..A, T,= 2flr..S.+ 2 ^.Ru". 



= 1 -! M-+l 



Consequently, if n > 0, 



|c.|s 2 \g.\A. m \ m - l r 



s 



r A \ 



-> 



i 

 -L 



1 J 



lA/t + By- 

 i.e., 



AG 

 ^'^Af -M Bv-' r ( 1 ) /> ' ' ' ....... ( 8A ) 



Also 



I " I ~ ^ I y<* \ I ^ I "I" ^ I i/mT^O* 1 I ) 



' = 1 m = + 1 



2 



" ' 





But, from the asymptotic expansion of the gamma function, 



where K' is finite and independent of n. 

 Hence 



IT^KIyG + GK'BA-y-'llX/n+l)^ ...... ( 8B ) 



Comparing (8x) and (SB) with (8), we see that *{/(*)} possesses an asymptotic 

 expansion with characteristics, /-, /, p, cr, valid for the range of values of x stated in 

 the enunciation. 



2 P 2 



