414 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Section Page 
92. Resemblance between an integral function and its derivative. The extension of Rolle’s 
Theorem. 495 
93. P p ' (z) is of the same order as P p (?), and an expression for its zeros can be theoretically 
obtained. Verification when p = 2.496 
94. The general theorem.497 
95. The nature of asymptotic expansions which involve a parameter.497 
96. Proof of § 94 in the case of a function of finite non-integral order greater than unity with 
algebraic zeros. 49S 
97. If F (?) be an integral function of finite order p with real zeros, F' (?) can at most have 
only p imaginary zeros.499 
98. The case when the zeros of F (?) all lie along any line through the origin.500 
99. Possibility of further applications.500 
Part I. 
Introduction. 
§ 1. Since the fundamental discoveries of Weierstrass, much progress has been 
made with regard to uniform transcendental functions : but the advances of modern 
u 
mathematics appear to have included no attempt formally to classify and investigate 
the properties of natural groups of such functions. 
Consider, for instance, the case of transcendental integral functions which admit 
one possible essential singularity at infinity. They form the most simple class of 
uniform functions of a single variable, and yet of them we know, broadly speaking, 
the nature of but four types :— 
(1) The exponential function, with which are associated circular and (rectangular) 
hyperbolic functions; 
(2) The gamma- functions ; 
(3) The elliptic functions and functions derived therefrom, such as the theta 
functions and Appeal's generalisation of the Eulerian functions ; 
(4) Certain functions which arise in physical problems (such as x~ n J : , (%)) whose 
properties have been extensively investigated for physical purposes. 
There are, of course, isolated examples of other types of functions ; yet, broadly 
speaking, except for algebraic polynomials, the four types just mentioned comprise 
the extent of our knowledge. 
§ 2. Take now an example of the first type of function. 
sinh 7r \/z ® 
We may write- 7 ^ = IT 
7 r V Z 
1 + 
?i = l [_ 
n- 
, and hence we have 
rr 
1 i. 
1 + 
r- — c "■ 
-7 
uu 
so that when \z\ is very large, the approximate value of II 
so long as — 7r < arg z < it. 
1 + 
'il¬ 
ls (27 t) 1 z - c~ ', 
