# §7.13 Zeros

## §7.13(i) Zeros of $\operatorname{erf}z$

$\operatorname{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $\mathbb{C}$ there is an infinite set of zeros $z_{n}=x_{n}+iy_{n}$, $n=1,2,3,\dots$, arranged in order of increasing absolute value. The other zeros of $\operatorname{erf}z$ 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 $n\to\infty$

 7.13.1 $\displaystyle x_{n}$ $\displaystyle\sim\lambda-\tfrac{1}{4}\mu\lambda^{-1}+\tfrac{1}{16}(1-\mu+% \tfrac{1}{2}\mu^{2})\lambda^{-3}-\cdots,$ $\displaystyle y_{n}$ $\displaystyle\sim\lambda+\tfrac{1}{4}\mu\lambda^{-1}+\tfrac{1}{16}(1-\mu+% \tfrac{1}{2}\mu^{2})\lambda^{-3}+\cdots,$ ⓘ 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 $\displaystyle\lambda$ $\displaystyle=\sqrt{(n-\tfrac{1}{8})\pi},$ $\displaystyle\mu$ $\displaystyle=\ln\left(\lambda\sqrt{2\pi}\right).$ ⓘ Defines: $\mu$ (locally) and $\lambda$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: 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

## §7.13(ii) Zeros of $\operatorname{erfc}z$

In the sector $\tfrac{1}{2}\pi<\operatorname{ph}z<\tfrac{3}{4}\pi$, $\operatorname{erfc}z$ has an infinite set of zeros $z_{n}=x_{n}+iy_{n}$, $n=1,2,3,\dots$, arranged in order of increasing absolute value. The other zeros of $\operatorname{erfc}z$ are $\overline{z}_{n}$. The zeros of $w\left(z\right)$ are $iz_{n}$ and $i\overline{z}_{n}$.

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 $n\to\infty$

 7.13.3 $\displaystyle x_{n}$ $\displaystyle\sim-\lambda+\tfrac{1}{4}\mu\lambda^{-1}-\tfrac{1}{16}(1-\mu+% \tfrac{1}{2}\mu^{2})\lambda^{-3}+\cdots,$ $\displaystyle y_{n}$ $\displaystyle\sim\lambda+\tfrac{1}{4}\mu\lambda^{-1}+\tfrac{1}{16}(1-\mu+% \tfrac{1}{2}\mu^{2})\lambda^{-3}+\cdots,$

where

 7.13.4 $\displaystyle\lambda$ $\displaystyle=\sqrt{(n-\tfrac{1}{8})\pi},$ $\displaystyle\mu$ $\displaystyle=\ln\left(2\lambda\sqrt{2\pi}\right).$ ⓘ Defines: $\lambda$ (locally) and $\mu$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: 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

## §7.13(iii) Zeros of the Fresnel Integrals

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 $\mathbb{C}$ $C\left(z\right)$ has an infinite set of zeros $z_{n}=x_{n}+iy_{n}$, $n=1,2,3,\dots$, 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 $-z_{n}$, $\overline{z}_{n}$, $-\overline{z}_{n}$, $iz_{n}$, $-iz_{n}$, $i\overline{z}_{n}$, $-i\overline{z}_{n}$ 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 $n\to\infty$ the $x_{n}$ and $y_{n}$ corresponding to the zeros of $C\left(z\right)$ satisfy

 7.13.5 $\displaystyle x_{n}$ $\displaystyle\sim\lambda+\frac{\alpha(\alpha\pi-4)}{8\pi\lambda^{3}}+\cdots,$ $\displaystyle y_{n}$ $\displaystyle\sim\frac{\alpha}{2\lambda}+\cdots$,

with

 7.13.6 $\displaystyle\lambda$ $\displaystyle=\sqrt{4n-1},$ $\displaystyle\alpha$ $\displaystyle=(2/\pi)\ln\left(\pi\lambda\right).$ ⓘ Defines: $\alpha$ (locally) and $\lambda$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: 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 $n\to\infty$ the $x_{n}$ and $y_{n}$ corresponding to the zeros of $S\left(z\right)$ satisfy (7.13.5) with

 7.13.7 $\displaystyle\lambda$ $\displaystyle=2\sqrt{n},$ $\displaystyle\alpha$ $\displaystyle=(2/\pi)\ln\left(\pi\lambda\right).$

## §7.13(iv) Zeros of $\mathcal{F}\left(z\right)$

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

For an asymptotic expansion of the zeros of $\int_{0}^{z}\exp\left(\tfrac{1}{2}\pi it^{2}\right)\,\mathrm{d}t$ ($=\mathcal{F}\left(0\right)-\mathcal{F}\left(z\right)$ $=C\left(z\right)+iS\left(z\right)$) see Tuẑilin (1971).