7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.11 Relations to Other Functions 7.13 Zeros

§7.12 Asymptotic Expansions

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Permalink:: http://dlmf.nist.gov/7.12
See also:: Annotations for Ch.7

§7.12(i) Complementary Error Function
§7.12(ii) Fresnel Integrals
§7.12(iii) Goodwin–Staton Integral

§7.12(i) Complementary Error Function

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Keywords:: Stokes phenomenon, asymptotic expansions, complementary error function, error functions, exponentially-improved
Referenced by:: §2.11(iv), §7.12(ii), Figure 7.3.6, Figure 7.3.6, Figure 7.3.6
Permalink:: http://dlmf.nist.gov/7.12.i
See also:: Annotations for §7.12 and Ch.7

As $z\to\infty$

7.12.1	$\displaystyle\operatorname{erfc}z$	$\displaystyle\sim\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{m=0}^{\infty}(-1)^{m}\frac% {{\left(\tfrac{1}{2}\right)_{m}}}{z^{2m+1}},$
7.12.1	$\displaystyle\operatorname{erfc}\left(-z\right)$	$\displaystyle\sim 2-\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{m=0}^{\infty}(-1)^{m}% \frac{{\left(\tfrac{1}{2}\right)_{m}}}{z^{2m+1}},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 7.1.23 (in different form) Referenced by: §3.5(ix), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/7.12.E1 Encodings: TeX, TeX, pMML, pMML, png, png Notational Change (effective with 1.0.9): Previously the RHS of these equations were written as $\frac{e^{-z^{2}}}{\sqrt{\pi}z}\sum_{m=0}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5% \cdots(2m-1)}{(2z^{2})^{m}}$ and $2-\frac{e^{-z^{2}}}{\sqrt{\pi}z}\sum_{m=0}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5% \cdots(2m-1)}{(2z^{2})^{m}}$ . We have rewritten these sums more concisely using Pochhammer’s symbol. Suggested 2014-03-13 by Giorgos Karagounis See also: Annotations for §7.12(i), §7.12 and Ch.7

both expansions being valid when $|\operatorname{ph}z|\leq\frac{3}{4}\pi-\delta$ ( $<\frac{3}{4}\pi$ ).

When $|\operatorname{ph}z|\leq\frac{1}{4}\pi$ the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when $\operatorname{ph}z=0$ . When $\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi$ the remainder terms are bounded in magnitude by $\csc\left(2|\operatorname{ph}z|\right)$ times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha=\frac{1}{2}$ and $z$ replaced by $z^{2}$ ; compare (7.11.2).

For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). (Note that some of these re-expansions themselves involve the complementary error function.)

§7.12(ii) Fresnel Integrals

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Keywords:: Fresnel integrals, asymptotic expansions, auxiliary functions for Fresnel integrals, exponentially-improved
Notes:: (7.12.2) and (7.12.3) follow from (7.7.10) and (7.7.11) by applying Watson’s lemma in its extended form (§2.4(i)). (7.12.4)–(7.12.7) follow from (7.7.10), (7.7.11), and the identity $(t^{2}+1)^{-1}=\sum_{m=0}^{n-1}(-1)^{m}t^{2m}+(-1)^{n}t^{2n}(t^{2}+1)^{-1}$ . The error bounds are obtained by setting $t=\sqrt{\tau}$ in (7.12.6) and (7.12.7), rotating the integration path in the $\tau$ -plane through an angle $-4\operatorname{ph}z$ , and then replacing $|\tau+1|$ by its minimum value on the path.
Permalink:: http://dlmf.nist.gov/7.12.ii
See also:: Annotations for §7.12 and Ch.7

The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and

7.12.2		$\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable A&S Ref: 7.3.27 Referenced by: §7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/7.12.E2 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): Previously the RHS of this equation was written as $\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(4m-1)}{(% \pi z^{2})^{2m}}$ . We have rewritten the sum more concisely using Pochhammer’s symbol. Suggested 2014-03-13 by Giorgos Karagounis See also: Annotations for §7.12(ii), §7.12 and Ch.7

7.12.3		$\mathrm{g}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},$
ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable A&S Ref: 7.3.28 Referenced by: §7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/7.12.E3 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): Previously the RHS of this equation was written as $\frac{1}{\pi^{2}z^{3}}\sum_{m=0}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(4% m+1)}{(\pi z^{2})^{2m}}$ We have rewritten the sum more concisely using Pochhammer’s symbol. Suggested 2014-03-13 by Giorgos Karagounis See also: Annotations for §7.12(ii), §7.12 and Ch.7

as $z\to\infty$ in $|\operatorname{ph}z|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)$ . The remainder terms are given by

7.12.4		$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\mathrm{f})}(z),$
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable, $n$ : nonnegative integer and $R_{n}^{(\mathrm{f})}(z)$ : remainder term A&S Ref: 7.3.27 (in different form) Referenced by: §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/7.12.E4 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): Previously the RHS of this equation was written as $\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{1\cdot 3\cdots(4m-1)}{(\pi z^{2})% ^{2m}}+R_{n}^{(\mathrm{f})}(z)$ . We have rewritten the sum more concisely using Pochhammer’s symbol. Suggested 2014-03-13 by Giorgos Karagounis See also: Annotations for §7.12(ii), §7.12 and Ch.7

7.12.5		$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\mathrm{g})}(z),$
ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable, $n$ : nonnegative integer and $R_{n}^{(\mathrm{g})}(z)$ : remainder term A&S Ref: 7.3.28 (in different form) Referenced by: Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/7.12.E5 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): Previously the RHS of this equation was written as $\frac{1}{\pi^{2}z^{3}}\sum_{m=0}^{n-1}(-1)^{m}\frac{1\cdot 3\cdots(4m+1)}{(\pi z% ^{2})^{2m}}+R_{n}^{(\mathrm{g})}(z)$ . We have rewritten the sum more concisely using Pochhammer’s symbol. Suggested 2014-03-13 by Giorgos Karagounis See also: Annotations for §7.12(ii), §7.12 and Ch.7

where, for $n=0,1,2,\dots$ and $|\operatorname{ph}z|<\tfrac{1}{4}\pi$ ,

7.12.6		$R_{n}^{(\mathrm{f})}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{% -\pi z^{2}t/2}t^{2n-(1/2)}}{t^{2}+1}\,\mathrm{d}t,$
ⓘ Defines: $R_{n}^{(\mathrm{f})}(z)$ : remainder term (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.3.29 (in different form) Referenced by: §7.12(ii) Permalink: http://dlmf.nist.gov/7.12.E6 Encodings: TeX, pMML, png See also: Annotations for §7.12(ii), §7.12 and Ch.7

7.12.7		$R_{n}^{(\mathrm{g})}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{% -\pi z^{2}t/2}t^{2n+(1/2)}}{t^{2}+1}\,\mathrm{d}t.$
ⓘ Defines: $R_{n}^{(\mathrm{g})}(z)$ : remainder term (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.3.30 (in different form) Referenced by: §7.12(ii) Permalink: http://dlmf.nist.gov/7.12.E7 Encodings: TeX, pMML, png See also: Annotations for §7.12(ii), §7.12 and Ch.7

When $|\operatorname{ph}z|\leq\frac{1}{8}\pi$ , $R_{n}^{(\mathrm{f})}(z)$ and $R_{n}^{(\mathrm{g})}(z)$ are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when $\operatorname{ph}z=0$ . They are bounded by $|\csc\left(4\operatorname{ph}z\right)|$ times the first neglected terms when $\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi$ .

For other phase ranges use (7.4.7) and (7.4.8). For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i).

§7.12(iii) Goodwin–Staton Integral

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Keywords:: Goodwin–Staton integral, asymptotic expansion
Permalink:: http://dlmf.nist.gov/7.12.iii
See also:: Annotations for §7.12 and Ch.7

See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$ .

7.11 Relations to Other Functions 7.13 Zeros

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§7.12 Asymptotic Expansions

Contents

§7.12(i) Complementary Error Function

§7.12(ii) Fresnel Integrals

§7.12(iii) Goodwin–Staton Integral