MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
493 
the function F (z) = IT 
n = 1 
1 + 
e n 
, which, by a change of the independent variable, 
reduces to Lambert’s function. 
For this function we have obtained the asymptotic expansion 
log F (z) = 1 (log zf ~ i log 2 + 
t r 2 * (- r 1 0, 
6 1 s i, S3' 
Therefore y log F (z) = ^ - 1 +5 ( —^ 
dz & v ’ z 2z T s =i z* +l 
The number of roots within a circle of radius r is, therefore, to terms which are 
exponentially small when r is large 
2-7TC 
log r 
e~' 6 + — {l 6 - i) re 10 idd = log r - 1. 
Since the function has no zero at the origin, we should ha.ve predicted the occurrence 
of the term — 
§ 87. We may now prove that, if the dominant term of the zero of an integral 
function is algebraic and such that the zero is of non-integral order. p (where p is 
neither zero nor infinite, but greater or less than unity), then the number of roots of 
the function within a circle of large radius r is to a first approximation 
sill 7 Tp 
IT 
log <f> (r), 
where <£ (r) is the maximum value of the modulus of the function on the circle in 
question. 
There are two cases to be considered according as p is greater or less than unity. 
We take the former, the argument will hold in detail for the latter by changing p 
into —. 
9 
Let F (z) be the function in question ; then, under the conditions enunciated, 
7T 
log F (z) is equal to . - z p - f- terms of lower order. 
Hence N, the number of roots required is, to a first approximation, given by 
N = 
r p e pL(> l dO = 
r p 
sin 7 rp 
IT 
log (r). 
We thus complete and prove Borel’s intuition. 
§ 88. When p is an integer, the preceding theorem ceases to be valid. But we can 
now prove that the number of roots to a first approximation is — ' . 
