Page:Proceedings of the Royal Society of London Vol 69.djvu/131

Rh

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