492 
MB, E. W. BARNES ON INTEGRAL FUNCTIONS. 
in which the terms neglected on the right-hand side are of lower exponential order 
than those retained. 
By Cauchy’s theorem the number of roots N within a circle of given large radius 
r is determined by 
N = - 1 ( y log T (z) dz, 
2m) dz & w ’ 
the integral being taken round the circle in question. 
Now we may, to terms which vanish exponentially with ^, substitute for T (z) its 
value given by the asymptotic expansion. And this expansion is valid for all values 
of z for which — n < arg z < v. It is also valid right up to the two limits of arg 2 , 
provided the circle on which z lies passes between two consecutive zeros of • 
If, now, 2 = r&°, we have, to terms which vanish exponentially with y > 
log r — lO — y + 
2 ( ~)'B, +1 c- 
t=o 2 ^ + 2 r -i+- 
re 
cie. 
Now 
ir u n a in ~ U' 
0 e‘ e cie = 
LIT 
LIT 
- 6<r‘ 6 
L 
7T 1 [n 
dO 
— r 
— i nel * + m 
0 —iir 1 
= V. 
Therefore, to terms which are ultimately exponentially small 
N = r -f i. 
Of course we know independently that 
the number of roots is the greatest integer 
less than r. And the entrance of the 
term \ might have been predicted a priori, 
for when the circle of radius r passes 
through a zero of ^ - we jump from — 
to + as we integrate round a small 
circle enclosing' this zero. 
8G. It is interesting to notice that the analysis verifies itself in the same way for 
