468 
ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
§ 63. The reader will notice that in the preceding analysis we have used the 
methods and not the result of the general formula. 
The reason is that with an exponential subject of integration we are unable to 
ensure that we do not introduce arbitrary additive constants when the indefinite 
integrals are transformed as formerly. 
For in this case <£ ( n ) = e n and i fj ( n ) = log n ; 
and we have to consider a series of integrals of which the first is 
<f> (m) 
p ( t ) dt 
We are tempted to say that this integral is equal to 
1 log [f (m) z 1 ] dt 
log (f) ( m) + lo. 
O’ t 
t 
whereas we only avoid introducing an additive constant by saying that 
f { r 3Ji = i [log 0 - (4 (m) t)T = J [log 4 (m)]l 
§ 64. The integral function just considered is the most simple function of zero 
order. In carrying out the algebraical analysis of a theory of such functions, it would 
be necessary to consider the types 
oo 
n 
n = l 
n 
n = 1 
l 
&c. 
The asymptotic expansions for these successive functions are of successively lower 
orders of greatness—they are never, however, of so low an order as z n , where n is 
finite. This agrees with the known theorem that an algebraical polynomial is the 
only uniforn function of such an order. Unfortunately, unless we introduce new 
analytical functions defined by definite integrals, we cannot investigate formally 
asymptotic approximations for such types; and until the properties of such new 
functions are investigated, we but express one unknown form in terms of another. 
Simple Integral Functions of Finite Non-integral Order Greater than Unity. 
§ 65. In the investigations to which we now proceed of simple integral functions of 
finite non-integral order greater than unity, the theoretical considerations which 
have been given in detail for functions of order less than unity will for the most part 
be suppressed, and for brevity only the bare analysis will be written down. 
We consider first the standard function Q p ( z ) = II ( 1 + Uyj e » 
where p > 1 and p is an integer such that p + 1 > p > p- 
Let 2 = He' 9 , and suppose that E. is very large. 
(Nf 
nil -PlP 
