384 
MR. E. BARNES ON THE THEORY OF THE 
where the quantity log 2 has its principal value with respect to the axis of — (wi + 
and Ju(z, Wo) is equal to the fundamental integral taken along a perpendicular 
contour cutting the axis of cr between —n and — (71 +1). It is evident that the 
integral when | 2 . | is large is of an order of magnitude less than . 
We therefore have the asymptotic expansion, when | 2 | is large, 
lO'" " 
og 
P: ("i, w.d 
= - 74 ; (a) [log 5 : - '2{m + m) ttl - | - 1] 
- .f, (^') [log z m) ttl - |] 
o ' / \ ri 0 / I o n I V 1 )AoS„;_^|(ft) 
— oS| (a) [log s — 2(?n + m) tti + N -;—=— 
and tlie residue after 7i terms of the final series have been taken is of the same order 
of mao-nitude as the final term taken. 
This expansion is evidently the same as that previously obtained. The limitation 
that a must be })Ositive with respect to the w’s may evidently be removed by 
employing the fundamental difference relations for the double gamma function and 
the asymptotic expansion for log E^ (2 + a). AVe are finally left with the essential 
limitation that s shall not lie within the barrier region hounded by the axis to — Wj 
and —Wo. 
The Tra7iscemlentally-t7'nnsceiideiital Nature of 1 ^( 2 ). 
§ 87. We finally pi'ove the theorem that the double gamma function cannot arise 
as the solution of a ditferential equation whose coefficients are not generated from the 
function itself Modifying slightly the nomenclature introduced by Mooee,^ we 
may say tliat rjj(z) is a transcendentally-transcendental function. The proof is a 
slight modification of that given for the G function (§ 30), which in turn was similar 
to the investigation of Part V. of the “ Theory of the Gamma Function.” 
In the first place it may he proved exactly as before that if the theorem is true for 
log To (2), it is true for ro( 2 ). We shall therefore confine ourselves to the considera¬ 
tion of the function 
0 (.) = - 7 log r, (z). 
By the fundamental ditterence equations of § 20, we have 
(2 + w,) - (h (z) = ili{z\ w^) 
</) (2 -h Wo) — (j> (2) = Xp{z\ W^) 
cl^ 
where, for convenience, we put x/j(z) = ri(2). 
* Moore, ‘Math. Ann.,’ vol. 48, pp. 49 et seq. MooRE uses the term only to describe functions which 
cannot be generated by a differential equation with algebraic coefficients. 
