MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
407 
If then we carry out the general process, we shall obtain the asymptotic expansion 
log II 
51 = 1 
1 + s 
— (m — 1) log z 
vi 
-o)+ S 
S — 1 
(-r 1 
sz“ 
i (-) 
8-1 
c. 
+ S' 
8= —Jc 
‘rill gins' 
I e ” s dn -— 
, m , 4 (-V : C S , 47 (-) s 1 (e ms c 
= (m - 1) log z - — + - + 2 ——— + X ——- ' - - 
Z L S=1 SZ s=-l- S* 
As before, we have to “sum” the final divergent series. We take \z\ to be a 
large quantity such that 
is very nearly equal to unity, and then we consider the 
X- /' Ns—1 f giOs piOs 
Fourier series S' - \ -— 
«=-* sl« 2 
But S' = 2 S ( — y \ C ° S - = - i (> - w )• 
s= —A- S- 3=1 S 
A ( _V t>‘ es 
And s' - = - ye. 
s=—A 2-S ^ 
Therefore we have the asymptotic expansion 
loff IT 
71=1 
1 + A 
/ im i m , 4 (-) s_l c, , , 
= (w - 1) log z - - + + s — yy— + 1 
S=1 
or finally* log n 
/i = i 
z 
1 ~h 
e“ 
= i ( lo g Z Y - 4 log 2 + ~ + s 
log 7 
(-r'c. 
O 0 
** 7T 
+ 7-Gog 
3=1 
62. It should be noticed that if, in the function whose asymptotic expansion has 
thus been obtained, we substitute e" for 2 , we shall obtain the function 14 
51=1 
This is an integral function whose zeros are of the form 
1 + «» 
z = n + (2m — 1) 
7TI 
n = 1, 2, 3,... 00 . 
7)1 — — OC , . . . , — 1, 0, 1, ... 00 
It is substantially what I propose to call Lambert’s function. The function has 
properties which are a sort of mean between those of the elliptic and double gamma 
functions. 
We can express Lambert’s function as a product of two double gamma functions. 
It is closely connected with the well-known Lambert’s series, and in terms of it we 
can express in a very elegant form the coefficients of capacity of two spheres. 
* The dominant terms of this result are equivalent to those given by MELLIN, ‘ Acta Soeietatis 
Fennicse,’ t. 24, p. 50. 
3 O 2 
