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

§7.12 Asymptotic Expansions

Permalink:: http://dlmf.nist.gov/7.12

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

§7.12(i) Complementary Error Function

Keywords:: Stokes phenomenon, complementary error function, error functions
Referenced by:: §2.11(iv), §7.12(ii), Figure 7.3.6, Figure 7.3.6
Permalink:: http://dlmf.nist.gov/7.12.i

As $z\to\infty$

7.12.1

$\mathop{\mathrm{erfc}\/}\nolimits z\sim\frac{e^{{-z^{2}}}}{\sqrt{\pi}z}\sum_{{% m=0}}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(2m-1)}{(2z^{2})^{m}},$

$\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z\right)\sim 2-\frac{e^{{-z^{2}}}}{% \sqrt{\pi}z}\sum_{{m=0}}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(2m-1)}{(2% z^{2})^{m}},$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : base of exponential function, $\sim$ : asymptotic equality and : complex variable
A&S Ref:: 7.1.23 (in different form)
Referenced by:: ¶ ‣ §3.5(ix)
Permalink:: http://dlmf.nist.gov/7.12.E1
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both expansions being valid when $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{3}{4}\pi-\delta$ ( $<\frac{3}{4}\pi$ ).

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

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 $-4\mathop{\mathrm{ph}\/}\nolimits z$ , and then replacing $|\tau+1|$ by its minimum value on the path.
Keywords:: Fresnel integrals, auxiliary functions for Fresnel integrals
Permalink:: http://dlmf.nist.gov/7.12.ii

The asymptotic expansions of $\mathop{C\/}\nolimits\!\left(z\right)$ and $\mathop{S\/}\nolimits\!\left(z\right)$ are given by (7.5.3), (7.5.4), and

7.12.2 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi z}\sum_{{m=0}}^% {\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(4m-1)}{(\pi z^{2})^{{2m}}},$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\sim$ : asymptotic equality and : complex variable
A&S Ref:: 7.3.27
Referenced by:: §7.12(ii), §7.12(ii)
Permalink:: http://dlmf.nist.gov/7.12.E2
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7.12.3 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi^{2}z^{3}}\sum_{% {m=0}}^{\infty}(-1)^{m}\frac{1\cdot 3\cdot 5\cdots(4m+1)}{(\pi z^{2})^{{2m}}},$

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\sim$ : asymptotic equality and : complex variable
A&S Ref:: 7.3.28
Referenced by:: §7.12(ii), §7.12(ii)
Permalink:: http://dlmf.nist.gov/7.12.E3
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as $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)$ . The remainder terms are given by

7.12.4 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\frac{1}{\pi z}\sum_{{m=0}}^{{n% -1}}(-1)^{m}\frac{1\cdot 3\cdots(4m-1)}{(\pi z^{2})^{{2m}}}+R_{n}^{{(\mathop{% \mathrm{f}\/}\nolimits)}}(z),$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, : complex variable and : nonnegative integer
A&S Ref:: 7.3.27 (in different form)
Referenced by:: §7.12(ii)
Permalink:: http://dlmf.nist.gov/7.12.E4
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7.12.5 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\frac{1}{\pi^{2}z^{3}}\sum_{{m=% 0}}^{{n-1}}(-1)^{m}\frac{1\cdot 3\cdots(4m+1)}{(\pi z^{2})^{{2m}}}+R_{n}^{{(% \mathop{\mathrm{g}\/}\nolimits)}}(z),$

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, : complex variable and : nonnegative integer
A&S Ref:: 7.3.28 (in different form)
Permalink:: http://dlmf.nist.gov/7.12.E5
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where, for $n=0,1,2,\dots$ and $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{4}\pi$ ,

7.12.6 $R_{n}^{{(\mathop{\mathrm{f}\/}\nolimits)}}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{e^{{-\pi z^{2}t/2}}t^{{2n-(1/2)}}}{t^{2}+1}dt,$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, : differential of , : base of exponential function, $\int$ : integral, : complex variable and : nonnegative integer
A&S Ref:: 7.3.29 (in different form)
Referenced by:: §7.12(ii)
Permalink:: http://dlmf.nist.gov/7.12.E6
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7.12.7 $R_{n}^{{(\mathop{\mathrm{g}\/}\nolimits)}}(z)=\frac{(-1)^{n}}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{e^{{-\pi z^{2}t/2}}t^{{2n+(1/2)}}}{t^{2}+1}dt.$

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, : differential of , : base of exponential function, $\int$ : integral, : complex variable and : 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

When $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{8}\pi$ , $R_{n}^{{(\mathop{\mathrm{f}\/}\nolimits)}}(z)$ and $R_{n}^{{(\mathop{\mathrm{g}\/}\nolimits)}}(z)$ 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 $\mathop{\mathrm{ph}\/}\nolimits z=0$ . They are bounded by $|\mathop{\csc\/}\nolimits\!\left(4\mathop{\mathrm{ph}\/}\nolimits z\right)|$ times the first neglected terms when $\frac{1}{8}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{4}\pi$ .

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

Keywords:: Goodwin–Staton integral
Permalink:: http://dlmf.nist.gov/7.12.iii

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

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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