### The Zeros of Certain Integral Functions

Quarterly Journal of Mathematics, Vol. 1 (1930), 38-59

Introduction (Excerpts)

I propose to consider functions of the form

(1.11) \(f(z) = f(x+iy) = f(re^{i\theta}) = \displaystyle \int_a^b e^{zt}\phi(t)\;dt\)

where \(\phi(t)\) is a complex function, integrable in the sense of Lebesgue. The functions

(1.12) \(U(t) = \displaystyle \int_a^b \cos ztu(t)\;dt\), \(V(z) = \displaystyle \int_a^b \sin ztv(t)\;dt\)

may be reduced to the form (1.11) by simple transformations. Among them are many well-known functions such as Bessel functions.

The problem here is to determine approximately the total number of zeros, and the position of the zeros for which *r* is large. Titchmarsh has proved a number of very general theorems of this character....

...

Pólya has proved a number of theorems of a different type concerning the zeros of the functions (1.11). In Pólya's theorems the hypotheses are much more restrictive, and the conclusions much more precise....

He also proves very precise results concerning the reality and location of the zeros of the functions (1.12); thus if *v*(t) is a positive increasing function, which is convex in the interval (0,1), and if *v*(t+0)=0, then all the zeros of
\[V(z) = \int_0^1 \sin ztv(t)\;dt \]
are real, and there is exactly one zero inside each of the intervals (π, 3π/2) (2π, 5π/2).... The well-known results for Bessel functions are included among these as special cases.

Very precise results have been obtained by Hardy, for particular functions, such as
\[f(z) = \int_0^1 e^{zt}t^{a-1}\left(\log\frac{1}{t}\right)^{s-1}dt = \sum_0^\infty \frac{\Gamma(s)z^n}{\Gamma(n+1)(a+n)^s}\]
The first new theorem of this paper, viz. VI, and the substance of VII are also due to Professor Hardy. I should like to thank him for permission to use his manuscript, and for many valuable suggestions and criticisms.

My object here is to prove a number of theorems intermediate in generality between those of Titchmarsh and Hardy. These theorems are of a rather different character from most of Pólya's, more general in some respects and less in others. I begin with a few general remarks by way of explanation.