## 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.