§7.13 Zeros

$\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. The other zeros of $\mathrm{erf}z$ are $-z_n$, ${\overline{z}}_n$, $-{\overline{z}}_n$.

Table 7.13.1 gives 10D values of the first five $x_n$ and $y_n$. For graphical illustration see Figure 7.3.5.

As $n\to \mathrm{\infty }$

where

In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. The other zeros of $\mathrm{erfc}z$ are ${\overline{z}}_n$. The zeros of $w\left(z\right)$ are $\mathrm{i}{z}_n$ and $\mathrm{i}{\overline{z}}_n$.

Table 7.13.2 gives 10D values of the first five $x_n$ and $y_n$. For graphical illustration see Figure 7.3.6.

As $n\to \mathrm{\infty }$

where

At $z=0$, $C\left(z\right)$ has a simple zero and $S\left(z\right)$ has a triple zero. In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. Similarly for $S\left(z\right)$. Let $z_n$ be a zero of one of the Fresnel integrals. Then $-z_n$, ${\overline{z}}_n$, $-{\overline{z}}_n$, $\mathrm{i}{z}_n$, $-\mathrm{i}{z}_n$, $\mathrm{i}{\overline{z}}_n$, $-\mathrm{i}{\overline{z}}_n$ are also zeros of the same integral.

Tables 7.13.3 and 7.13.4 give 10D values of the first five $x_n$ and $y_n$ of $C\left(z\right)$ and $S\left(z\right)$, respectively.

As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $C\left(z\right)$ satisfy

with

As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $S\left(z\right)$ satisfy (7.13.5) with

In consequence of (7.5.5) and (7.5.10), zeros of $ℱ\left(z\right)$ are related to zeros of $\mathrm{erfc}z$. Thus if $z_n$ is a zero of $\mathrm{erfc}z$ (§7.13(ii)), then $(1+\mathrm{i})z_n/\sqrt{\pi }$ is a zero of $ℱ\left(z\right)$.

For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971).