ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
427 
In particular, we may raise an asymptotic series to any finite power, and it will 
then represent the corresponding power of the function represented by the original 
series. 
The term-by-term integral of an asymptotic series is equal to the integral of the 
function which it represents : in brief, we may integrate an asymptotic series. 
In general, we may not differentiate an asymptotic equality. 
[Nevertheless, we may differentiate most of the expansions which arise naturally 
in analysis, and are not constructed artificially.] 
Similarly, if an asymptotic equality involves an arbitrary parameter, we may not 
in general (but we may fairly safely in practice) differentiate with respect to that 
parameter. 
Such are the main propositions of the arithmetic theory of asymptotic expansions. 
The difficulties inherent in the theory are obvious when we attempt its application. 
We have, in all cases, to investigate a superior limit to the remainder of the series 
after the first (n + 1) terms have been taken ; and, to do this, we must have 
command, even for the most simple cases, of analytical processes of great complexity 
and power. 
§ 24. We proceed then to consider these series from the function-theoretic point of 
view. 
That is to say, on the one hand, we attempt to oive a definition to a divergent 
series which shall harmonise with the development of Weiersteass’ theory, and on 
the other, we enter more deeply into the nature of the essential singularity of the 
function of which the divergent series is the expansion. 
Suppose-first that we have a series ci 0 + aq z fi- . . . + a IL z n + ... of finite radius 
of convergency p , so that by Caijchy’s rule, L t '{/ a ,, = p ~ Y . 
n =oo 
When \ z\ is greater than p, the series is divergent and our fundamental conception 
of a series as a command to add in order successive terms leads to no result. 
And yet, if the function which the series represents be not one which has the circle 
of radius p as a line of essential singularity, the function exists outside this circle, 
and admits an analytic continuation. Thus the function exists even when the series 
is divergent. 
Can we not then regard the series when divergent as a command to perform certain 
operations which shall yield the analytic continuation of the function? We can do 
so, and in an infinite number of ways. 
The most simple is, perhaps, given by an extension of a process developed by 
BorelA 
Let the plane of the variable x be dissected by some line going from 0 to oo to the 
right of the axis of y. 
* “Th^orie des series divergentes sommables,” ‘ Liouville,’ 5 ser., t. 2, pp. 103 et seq. “ M4moire snr les 
series divergentes,” Ann. de l’Ecole Normale Superieure, 3 ser., t. 16, pp. 1 et seq. 
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