372 
MR. E. W. BARXES OX THE THEORY OF THE 
Part V. 
The Asymptotic Expansion and Transcendentadly-transcendental Nature of the 
Double Gamma Function. 
§ 78. There remains now the consideration of two more general characteristics of 
the double gamma function :— 
1. It admits over part of the region near infinity an asymptotic ex 2 :)ansion in powers 
of the variable. 
2. It cannot be obtained as the solution of a differential equation whose coefficients 
involve exclusively more simple functions. 
It will afterwards be seen that these characteristics are common to all gamma 
functions. 
Let us consider first the behaviour of V^{z) near infinity. We know that its poles 
are given by 
m. = 0, 1, . . . oo 1 
2 + mpj)y + m.-yO}.-, = 0 I ’ ’ L 
nio = 0, 1, . . . CO J 
-Ci)2 
Therefore near z — ^ the jDoles of To {z) are massed together between and on the 
negative axes of ojj and co.,, so as to form a lacunary space on the equivalent portion 
of the Neumann sphere. Between these axes, therefore, an asymptotic expansion 
cannot represent the function. We have to consider whether such an expansion can 
exist outside this lacunary area, within, that is to say, the non-shaded portion of the 
figure. 
We shall in the first place proceed entirely algebraically. It will be proved that 
within this non-shaded area a cpiasi-Laurent asymptotic expansion of the form 
{l,2)Oog3 + (l.a^+ 5 J 
?• = 1 
