Agnes Scott College

Mary L. Cartwright

The Zeros of Certain Integral Functions (II)
Quarterly Journal of Mathematics, Vol. 2 (1931), 113-129

Introduction (Excerpts)

In a previous paper I considered functions of the form \[f(z) = f(x+iy) = f(re^{i\theta}) = \int_{-1}^1 e^{zt}\phi(t)dt\]

where \(\phi(t)\) is a complex function, integrable in the sense of Lebesgue, and \(\phi(t)\) tends to a finite limit other than 0 at each end. The object of this paper is to consider the cases in which \(\phi(t)\) tends to 0 or ∞ at one or both ends. If \(\phi(t) \rightarrow \infty\) as \(t \rightarrow \pm 1\), we need lighter restrictions in order to obtain equivalent results, and if \(\phi(t) \rightarrow 0\), heavier ones, as the following example shows.