A Memoir on Integral Function*. 123 



other hand, the series for those values of z for which it is divergent 

 is not the arithmetically asymptotic expansion of the function. 



At this stage it is natural to invent some extension of the ideas thus 

 employed, and to apply that extension to the case in which the series 

 has zero radius of convergence. It is shown that this new problem is 

 radically different from the old one, the number of solutions being 

 infinite. The solutions given by the process of integration are, how- 

 ever, functions which have zero for their essential singularity, and 

 which admit the given series as an arithmetically asymptotic expansion 

 in certain regions whose apex is that singularity. 



We classify divergent series according to their order ; a series is said 



to be of order k when '-" is finite and not zero. 

 n k 



Series of the first order can be " summed " by the same process as 

 that employed for series of finite radius of convergence. Series of 

 higher order can be " summed " either by successive repetition of this 

 process, or by means of auxiliary functions derived from the higher 

 hypergeometric functions. 



An application is made to the Maclaurin sum formula, and it is 

 shown that this formula may be used to give an asymptotic expansion 



,/t-i 

 for 2 < (M), when m is large, when $ (z) is an integral function of z 



n=i 



of order greater than or equal to unity. 



Finally, the rearrangement of asymptotic series is considered, and 

 Part II closes with a theoretical account of the possibility and nature 

 of the asymptotic expansion of an integral function near its essential 

 singularity. 



, In Part III actual asymptotic expansions of wide classes of simple 

 integral functions with non-repeated zeros are obtained. 



The three standard functions employed are 



, where 



where p is greater than unity and not integral, and p is an integer such 

 that p + 1 > p > p ; 



where p is an integer 



