ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
499' 
differ from the corresponding zero of the original function by the term--y—y > 
P 
together with terms involving lower powers of n. 
In an exactly similar manner it may be proved that the function R' p (2) admits 
an asynrptotic expansion of which the dominant term is 
(-) p pz '- 2)0+p - 3/2 exp. j (-) p - 1 - p + log log z - i log 2 tt + ( J 8 F j , 
so that R' p (2) is of integral order p. 
The term log log 2 in the exponential just written down shows that we shall come, 
sooner or later, to a transcendental term in the expansion of the n ,h zero of R' p (2). 
Similarly the theorem may be established for the general simple non-repeated 
function of finite integral order. 
As regards the application of the same methods to simple repeated functions it is 
only necessary to notice that corresponding to a zero k times repeated of the original 
function there will be a zero (k — 1) times repeated of its derivative. 
§ 97 . We have now to consider whether the derivative of an integral function, all 
of whose roots are real, can have zeros other than the real zeros which by the 
extension of Rolle’s theorem separate the roots of the original function. 
For this purpose let us consider the difference between the number of roots of P (2) 
and of V (2) within a circle of very large radius r. 
This number will be N = ~||y logP'fz) — ; log P(z)| dz 
1 c d p' (z) 
= -— j - log — . dz, where the integral is taken round the 
circle in question. 
Now by examining the various cases which can arise, it may at once be seen 
d Y (z) 
that the asymptotic expansion of y log " is given by (p — l)logz + terms which 
_L l £ ) 
vanish when \z\ = x>. Therefore to a first approximation we have N = p — 1. 
If then the function P (2) is of genre p, its derivative can at most have only p zeros 
besides those demanded by the extension of Rolle’s theorem. 
And therefore when p is odd, P' (2) can at most have only (p — 1) imaginary roots.* 
In particular when P (2) is of genre 0 or 1, P' (2) can have no imaginary roots.! 
From this theorem coupled with the expansion given in § 5 and the equality 
d J n (2) _ (2) 
dz z n z' 1 
* Bokel, ‘ Fonctions Entieres,’ p. 44. 
t Laguerre, ‘ G^uvres,’ t. 1, pp. 167 et seq. 
