### On the Asymptotic Periods of Integral Functions

Proceedings of the Cambridge Philosophical Society, Vol. 31 (1935), 543-554

Communicated by Miss M. L. Cartwright

Received 15 March, read 28 October 1935

Introduction

A period of a function f(z) is defined to be a number ω (≠0) such that
\[\Delta_\omega(z) = f(z+w)-f(z)\]
is identically zero; and it can be shown that an integral function may either have no periods or else a single sequence \(k\lambda (k=\pm1,\pm2,\dots).\)

J.M. Whittaker defined an *asymptotic period* of an integral or meromorphic function as a number β (≠0) such that \(\Delta_\beta(z)\) is of lower order than f(z), and proved that an integral function \(f(z)\) has no asymptotic periods if
\[\overline{\lim_{r \rightarrow \infty}}\frac{\log M(r)}{r}=0\]
while if
\[\lim_{r \rightarrow \infty} \frac{(\log r)^2 \log M(r)}{r} = 0\]
there are no asymptotic periods β such that \(\Delta_\beta(z)\) is of order < 1. He has since shown that a function of order one either has no asymptotic periods or else a single sequence \(k\lambda (k=\pm1,\pm2,\dots).\) A function of any order greater than one can have a non-denumerable set of asymptotic periods.

He also proved that the set of all asymptotic periods of an integral function is linear and of measure zero. An important part in the proof is played by an extension of Guichard's theorem that every integral function has a finite sum, namely

Theorem A. *If f(z) is an integral function of finite order ρ, there is an integral function g(z) of order ρ for which \(g(z+\omega)-g(z)=f(z).\)*

I shall call numbers β satisfying the above definition *asymptotic periods of first kind*, and extend the definition as follows:

Def. *A number β* (≠0) *is an asymptotic period of an integral or meromorphic function f(z) if Δ*_{β}(z) is of lower order, or of the same order and lower type than f(z).

The set of all asymptotic periods of f(z) is denoted by B.

It is possible to extend Guichard's theorem further by considering the type of \(f(z)\) (see Theorem I), and hence to prove that B is a linear set of measure zero.

I also prove, using Theorem I and Carlson's theorem, that \(f(z)\) has no asymptotic periods if
\[\overline{\lim_{r \rightarrow \infty}}\frac{\log M(r)}{r} = 0,\]
while, if
\[\overline{\lim_{r \rightarrow \infty}}\frac{\log M(r)}{r} = \alpha \quad\quad (0 \lt \alpha \lt \infty)\]
there are no asymptotic periods of magnitude less than π/α.