12-1 A Memoir on Integral Functions. 



It is shown that, asymptotically near ,? = oo , we have 



and 



log R p (.:') = ( - ,r)p log z + ( - K 1 -$ log z - ~ log 2 



F (*//>) 



+ 2' 

 =-P 



where the numbers F are substantially Biemann {-functions of nega- 

 tive argument. 



It is then shown that these expansions can be generalised so as to 

 apply to wide classes of functions, notably those with algebraic 

 sequence of zeros. General formulae are given which symbolically 

 contain the expansion of all integral functions of the type considered, 

 which are such that the Maclaurin sum formula can be applied to the 

 function expressing the dependence of the nth zero upon n. And as 

 an example of a function with transcendental sequence, the asymptotic 

 expansion of 



00 r 



n 



= i L 



1+- 



is obtained completely. 



Part IV deals with the asymptotic expansion of repeated integral 

 functions of simple sequence. It is necessary to take an extended 

 definition of " order " in the case of such functions, and then actual 

 expansions are obtained for standard functions 



(1.) Of finite (non-zero or zero) order less than unity ; 



(2.) Of finite non-integral order greater than unity ; 



(3.) Of finite integral order greater than or equal to unity. 



Subsequently symbolic formulae are given for all integral functions 

 of the prescribed type ; and, as an example, the asymptotic expansion 

 of a repeated function with a transcendental index is considered. The 

 formulae are verified in the case of the G-function. 



Part V is devoted to applications of the previous expansions to the 

 problems mentioned in the introduction. 



A knowledge of the asymptotic expansion of a function serves to 

 determine the number of roots which it possesses inside a circle of 

 given large radius. If the function is of order p, the number of roots 

 within a circle of large radius r is to a first approximation 



sin Trp 



