7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.12 Asymptotic Expansions 7.14 Integrals

§7.13 Zeros

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Permalink:: http://dlmf.nist.gov/7.13
See also:: Annotations for Ch.7

§7.13(i) Zeros of $\operatorname{erf}z$
§7.13(ii) Zeros of $\operatorname{erfc}z$
§7.13(iii) Zeros of the Fresnel Integrals
§7.13(iv) Zeros of $\mathcal{F}\left(z\right)$

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

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Keywords:: asymptotic expansions, error functions, zeros
Notes:: See Fettis et al. (1973).
Referenced by:: Figure 7.3.5, Figure 7.3.5, Figure 7.3.5
Permalink:: http://dlmf.nist.gov/7.13.i
See also:: Annotations for §7.13 and Ch.7

$\operatorname{erf}z$ has a simple zero at , 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.

Table 7.13.1: Zeros $x_{n}+iy_{n}$ of $\operatorname{erf}z.$

	$x_{n}$	$y_{n}$
1	1.45061 61632	1.88094 30002
2	2.24465 92738	2.61657 51407
3	2.83974 10469	3.17562 80996
4	3.33546 07354	3.64617 43764
5	3.76900 55670	4.06069 72339

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Symbols:: $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{i}$ : imaginary unit, : complex variable, : nonnegative integer, $x_{n}$ : realpart of zero and $y_{n}$ : imagpart of zero
Keywords:: error functions, tables, zeros
A&S Ref:: Table 7.10 (in slightly different form)
Referenced by:: §7.13(i)
Permalink:: http://dlmf.nist.gov/7.13.T1
See also:: Annotations for §7.13(i), §7.13 and Ch.7

As $n\to\infty$

7.13.1

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

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

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : 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

$\lambda=\sqrt{(n-\tfrac{1}{8})\pi},$

$\mu=\ln\left(\lambda\sqrt{2\pi}\right).$

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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 : 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 $\operatorname{erfc}z$

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Keywords:: error functions, zeros
Notes:: See Fettis et al. (1973) and Elbert and Laforgia (2008).
Referenced by:: §12.11(ii), §7.13(iv), Figure 7.3.6, Figure 7.3.6, Figure 7.3.6, Erratum (V1.0.5) for References, Erratum (V1.0.5) for Clarifications
Permalink:: http://dlmf.nist.gov/7.13.ii
Addition (effective with 1.0.5):: Elbert and Laforgia (2008) has been added as a reference for this subsection. Also, the first sentence was stated more precisely by revising the leading clause.
See also:: Annotations for §7.13 and Ch.7

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.

Table 7.13.2: Zeros $x_{n}+iy_{n}$ of $\operatorname{erfc}z.$

	$x_{n}$	$y_{n}$
1	−1.35481 01281	1.99146 68428
2	−2.17704 49061	2.69114 90243
3	−2.78438 76132	3.23533 08684
4	−3.28741 07894	3.69730 97025
5	−3.72594 87194	4.10610 72847

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Symbols:: $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{i}$ : imaginary unit, : complex variable, : nonnegative integer, $x_{n}$ : realpart of zero and $y_{n}$ : imagpart of zero
Keywords:: error functions, tables, zeros
Referenced by:: §7.13(ii)
Permalink:: http://dlmf.nist.gov/7.13.T2
See also:: Annotations for §7.13(ii), §7.13 and Ch.7

As $n\to\infty$

7.13.3

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

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

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Symbols:: $\sim$ : Poincaré asymptotic expansion, : 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

$\lambda=\sqrt{(n-\tfrac{1}{8})\pi},$

$\mu=\ln\left(2\lambda\sqrt{2\pi}\right).$

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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 : 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

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Keywords:: Fresnel integrals, asymptotic expansions, zeros
Notes:: See Kreyszig (1957) and Fettis and Caslin (1973).
Permalink:: http://dlmf.nist.gov/7.13.iii
See also:: Annotations for §7.13 and Ch.7

At , $C\left(z\right)$ 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.

Table 7.13.3: Complex zeros $x_{n}+iy_{n}$ of $C\left(z\right).$

	$x_{n}$	$y_{n}$
1	1.74366 74862	0.30573 50636
2	2.65145 95973	0.25290 39555
3	3.32035 93363	0.22395 34581
4	3.87573 44884	0.20474 74706
5	4.36106 35170	0.19066 97324

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Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\mathrm{i}$ : imaginary unit, : real variable, : complex variable and : nonnegative integer
Keywords:: Fresnel integrals, tables, zeros
A&S Ref:: Table 7.11 (modification of)
Referenced by:: §7.13(iii)
Permalink:: http://dlmf.nist.gov/7.13.T3
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 $C\left(z\right)$ satisfy

7.13.5

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

$y_{n}\sim\frac{\alpha}{2\lambda}+\cdots$ ,

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : real variable, : 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

$\lambda=\sqrt{4n-1},$

$\alpha=(2/\pi)\ln\left(\pi\lambda\right).$

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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, : real variable and : 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

Table 7.13.4: Complex zeros $x_{n}+iy_{n}$ of $S\left(z\right)$ .

	$x_{n}$	$y_{n}$
1	2.00925 70118	0.28854 78973
2	2.83347 72325	0.24428 52408
3	3.46753 30835	0.21849 26805
4	4.00257 82433	0.20085 10251
5	4.47418 92952	0.18768 85891

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Symbols:: $S\left(\NVar{z}\right)$ : Fresnel integral, $\mathrm{i}$ : imaginary unit, : real variable, : complex variable and : nonnegative integer
Keywords:: Fresnel integrals, tables, zeros
A&S Ref:: Table 7.11 (modification of)
Referenced by:: §7.13(iii)
Permalink:: http://dlmf.nist.gov/7.13.T4
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

$\lambda=2\sqrt{n},$

$\alpha=(2/\pi)\ln\left(\pi\lambda\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : 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

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

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Keywords:: Fresnel integrals, asymptotic expansions, zeros
Permalink:: http://dlmf.nist.gov/7.13.iv
See also:: Annotations for §7.13 and Ch.7

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).

7.12 Asymptotic Expansions 7.14 Integrals

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§7.13 Zeros

Contents

§7.13(i) Zeros of

§7.13(ii) Zeros of

§7.13(iii) Zeros of the Fresnel Integrals

§7.13(iv) Zeros of

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

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

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