Zeros of sections of exponential integrals

For complex $z$, define the function \[ F_a(z) = \int_{a-1}^a \left(t-\frac{1}{2}\right)^2 e^{zt}\,dt \] and let \[ s_n[F_a](z) = \sum_{k=0}^{n} \frac{z^k}{k!} \int_{a-1}^a \left(t-\frac{1}{2}\right)^2 t^k\,dt \] be the $n^{\text{th}}$ partial sum of its Maclaurin series. We refer to $s_n[F_a](z)$ as the $n^{\text{th}}$ section of the exponential integral $F_a(z)$.

As $n \to \infty$, most zeros of the sections $s_n[F_a](z)$ grow on the order of $O(n)$. Consequently we rescale, replacing $z$ by $nz$, and instead consider the zeros of the normalized sections $s_n[F_a](nz)$. The complex zeros of these normalized sections accumulate on the curve \[ \begin{align*} D(a) &= \left\{z \in \mathbb{C} \,\colon \Re(z) \leq 0,\,\,\, |z| \leq \frac{2}{|1-2a|+1},\,\,\, \text{and } \left|ze^{1-(1-a)z}\right| = \frac{2}{|1-2a|+1}\right\} \\ &\qquad \cup \,\left\{z \in \mathbb{C} \,\colon \Re(z) \geq 0,\,\,\, |z| \leq \frac{2}{|1-2a|+1},\,\,\, \text{and } \left|ze^{1-az}\right| = \frac{2}{|1-2a|+1}\right\} \\ &\qquad \cup \,\left\{z \in \mathbb{C} \,\colon \Re(z) = 0 \,\,\,\text{and}\,\,\, |z| \leq \frac{2}{(|1-2a|+1)e} \right\}, \end{align*} \] which consists of two bulbous components, one in each half-plane, and a line segment on the imaginary axis. This curve $D(a)$ is called the Szegő curve for the function $F_a$.

Below is an image which shows the zeros of the first $40$ normalized sections (that is, the zeros of $s_n[F_a](nz)$ for $n=1,2,\ldots,40$) and their Szegő curve $D(a)$. The parameter $a$ ranges from $0$ to $1$ and back to $0$ in the animation. If we were to plot the zeros of more sections they would cluster nearer and nearer to the Szegő curve.

For more information about this plot and the information above you can check out my master's thesis and this paper.

Antonio R. Vargas

February 18, 2013