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



and cr, the greater of the quantities ^ and \a\ +p+ P \ay where p., ? 

 greatest possible values of \ x (*+ a)' 1 | , | (*+ a)-'| respectively for the values of x unde, 

 consideration. 



It is easily shown that 



Differentiating r times with respect to a, we get 



- 



where 



d r I a 



But, by LEIBNIZ' theorem, 



d r I a 



tt r 12 T 



the quantities ,C,, A, ... being the binomial coefficients. 

 Consequently 



d' I a" \l nl| 



o|"- r f, 

 . (n-r) '.[ 



x+a 



a 



.x + a 





so that 



' r+1 ' * |z+a|.(n-r) 

 Now f(x+a) possesses an expansion of the form 



nr -fi + JLT. 



|.(n r)!L |* + |. 



where | a, | Ap"F (i- + 1), 



Substituting for the negative powers of x + a from (10), we get 



f(x+a) = l <> +^+ ... 



where 



a; 



6. = a.-.-A . a . a.-i+.-iC, . a 2 . a,_ 3 - . . . 



