419 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
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The memoir deals almost exclusively with simple integral functions of finite or zero 
order (vide the definitions of the succeeding paragraphs.) 
I reserve the consideration of functions of infinite order, and also the results which 
I have obtained in connection with functions of double or multiple sequence. The 
latter form a self-contained theory, which is a natural extension of the investigations 
of the present memoir. The consideration of the asymptotic expansion of integral 
functions defined by a Taylor’s series is also postponed, although certain noteworthy 
extensions of Hadamard’s results can be at once deduced from the present theory. # 
My thanks are due to Professor Forsyth for the kind way in which he has 
supplied me with references and criticism. 
The Classification of Integral Functions. 
§11. An integral function we define to be a uniform transcendental function with 
no poles, and a single essential singularity at infinity. [Sometimes it is convenient 
to include algebraic polynomials.] An integral function is thus the same as a 
holomorphic function, to use the translation of Cauchy’s name; it is the equivalent 
of the French “ fonction entieref and the German “ganze Funhtion.” Every mero- 
morphic function can be expressed as the quotient of two integral functions. 
The most simple integral function can be written in the form 
e K(z) n 
71 = 1 
/ 
where H (z) is an integral function of 2 , where the zero a n depends solely upon n and 
certain definite constants, and where the law of dependence of a n upon n is the 
same for all zeros. Such a function we call a simple integral function with a single 
sequence of non-repeated zeros. The law of dependence may be broken for a finite 
number of arbitrary zeros in the finite part of the plane. The existence of such zeros 
is equivalent to the multiplication of the transcendental function by an arbitrary 
polynomial coupled possibly with an exponential function of the type e p(z) , where 
p ( 2 ) is another algebraic polynomial. Such terms do not substantially alter the 
character of the function. 
Functions of the type c H(z) , where H ( 2 ) is an integral function, belong to a class 
apart. The integral function which we consider we shall suppose to be deprived of 
such extraneous factor. 
\ P T 1 A (±\ 
_,m=l ™ 
* The present memoir was largely written during the summer of the.year 1898. In consequence, and 
in spite of rigorous revision, results may sometimes appear to be tacitly claimed as new which have since 
been published in papers to which reference is made in connection with other investigations of the 
memoir. 
3 H 2 
