§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 $n\to \mathrm{\infty }$

7.13.1		$x_n$	$\sim \lambda -\frac{1}{4}\mu {\lambda }^{-1}+\frac{1}{16}(1-\mu +\frac{1}{2}{\mu }^2){\lambda }^{-3}-\mathrm{\cdots },$
		$y_n$	$\sim \lambda +\frac{1}{4}\mu {\lambda }^{-1}+\frac{1}{16}(1-\mu +\frac{1}{2}{\mu }^2){\lambda }^{-3}+\mathrm{\cdots },$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer, $x_n$: realpart of zero, $y_n$: imagpart of zero, $\mu$ and $\lambda$ A&S Ref: Table 7.10 (has two-term approximation) Permalink: http://dlmf.nist.gov/7.13.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(i), §7.13 and Ch.7

where

7.13.2		$\lambda$	$=\sqrt{(n-\frac{1}{8})\pi },$
		$\mu$	$=\ln \left(\lambda \sqrt{2\pi }\right).$
ⓘ Defines: $\mu$ (locally) and $\lambda$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/7.13.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(i), §7.13 and Ch.7

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 $n\to \mathrm{\infty }$

7.13.3		$x_n$	$\sim -\lambda +\frac{1}{4}\mu {\lambda }^{-1}-\frac{1}{16}(1-\mu +\frac{1}{2}{\mu }^2){\lambda }^{-3}+\mathrm{\cdots },$
		$y_n$	$\sim \lambda +\frac{1}{4}\mu {\lambda }^{-1}+\frac{1}{16}(1-\mu +\frac{1}{2}{\mu }^2){\lambda }^{-3}+\mathrm{\cdots },$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer, $\lambda$, $x_n$: realpart of zero, $y_n$: imagpart of zero and $\mu$ Permalink: http://dlmf.nist.gov/7.13.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(ii), §7.13 and Ch.7

where

7.13.4		$\lambda$	$=\sqrt{(n-\frac{1}{8})\pi },$
		$\mu$	$=\ln \left(2\lambda \sqrt{2\pi }\right).$
ⓘ Defines: $\lambda$ (locally) and $\mu$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/7.13.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(ii), §7.13 and Ch.7

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

7.13.5		$x_n$	$\sim \lambda +{\displaystyle \frac{\alpha (\alpha \pi -4)}{8\pi {\lambda }^3}}+\mathrm{\cdots },$
		$y_n$	$\sim {\displaystyle \frac{\alpha }{2\lambda }}+\mathrm{\cdots }$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $n$: nonnegative integer, $\alpha$ and $\lambda$ Referenced by: §7.13(iii) Permalink: http://dlmf.nist.gov/7.13.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(iii), §7.13 and Ch.7

with

7.13.6		$\lambda$	$=\sqrt{4n-1},$
		$\alpha$	$=(2/\pi )\ln \left(\pi \lambda \right).$
ⓘ Defines: $\alpha$ (locally) and $\lambda$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer A&S Ref: Table 7.11 (has slightly different form for $x_n$; for this correction see Fettis and Caslin (1973)) Permalink: http://dlmf.nist.gov/7.13.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(iii), §7.13 and Ch.7

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

7.13.7		$\lambda$	$=2\sqrt{n},$
		$\alpha$	$=(2/\pi )\ln \left(\pi \lambda \right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $\alpha$ and $\lambda$ Permalink: http://dlmf.nist.gov/7.13.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.13(iii), §7.13 and Ch.7

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).