7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.4 Symmetry7.6 Series Expansions

§7.5 Interrelations

ⓘ

Keywords:

Dawson’s integral, Fresnel integrals, Goodwin–Staton integral, auxiliary functions, error functions, interrelations, relation to error functions, relations to Dawson’s integral and exponential integral, relations to other functions

Notes:

(7.5.1) follows from (7.2.1)–(7.2.3). (7.5.2) follows from (7.2.6)–(7.2.9). (7.5.3) and (7.5.4) follow from (7.2.10) and (7.2.11). (7.5.5) and (7.5.6) follow from (7.2.10), (7.2.11), and (7.5.2). (7.5.8) follows from (7.2.1), (7.2.7), and (7.2.8). (7.5.9) follows from (7.2.2), (7.2.3), (7.5.7), and (7.5.8). (7.5.10) follows from (7.2.2), (7.5.6), and (7.5.8). (7.5.11) follows from (7.5.5). (7.5.12) follows from (7.4.8) and (7.5.11). For (7.5.13) see Olver (1997b, p. 44).

Permalink:

http://dlmf.nist.gov/7.5

See also:

Annotations for Ch.7

7.5.1		$$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$
ⓘ Symbols: $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E1 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.2		$$C\left(z\right)+\mathrm{i}S\left(z\right)=\frac{1}{2}(1+\mathrm{i})-ℱ\left(z\right).$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $ℱ\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E2 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.3		$$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable A&S Ref: 7.3.9 Referenced by: §7.12(ii), §7.5 Permalink: http://dlmf.nist.gov/7.5.E3 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.4		$$S\left(z\right)=\frac{1}{2}-\mathrm{f}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right).$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable A&S Ref: 7.3.10 Referenced by: §7.12(ii), §7.5 Permalink: http://dlmf.nist.gov/7.5.E4 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.5		$${\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}{z}^2}ℱ\left(z\right)=\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right).$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.13(iv), §7.5, §7.7(ii) Permalink: http://dlmf.nist.gov/7.5.E5 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.6		$${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E6 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

In (7.5.8)–(7.5.10)

7.5.7		$$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$
ⓘ Defines: $\zeta$: change of variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.12(ii), §7.5 Permalink: http://dlmf.nist.gov/7.5.E7 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

and either all upper signs or all lower signs are taken throughout.

7.5.8		$$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable Referenced by: §7.5, §7.5, §7.6(i) Permalink: http://dlmf.nist.gov/7.5.E8 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.9		$$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E9 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.10		$$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable A&S Ref: 7.3.22 (in different form) Referenced by: §7.12(ii), §7.13(iv), §7.5, §7.5 Permalink: http://dlmf.nist.gov/7.5.E10 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.11		$${|ℱ\left(x\right)|}^2={\mathrm{f}}^2\left(x\right)+{\mathrm{g}}^2\left(x\right),$$
$x\ge 0$,
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral and $x$: real variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E11 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

7.5.12		$${|ℱ\left(x\right)|}^2=\begin{array}{l}2+{\mathrm{f}}^2\left(-x\right)+{\mathrm{g}}^2\left(-x\right)-2\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi x^2\right)\mathrm{f}\left(-x\right)\\ -2\sqrt{2}\cos \left(\frac{1}{4}\pi -\frac{1}{2}\pi x^2\right)\mathrm{g}\left(-x\right),\end{array}$$
$x\le 0$.
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $x$: real variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E12 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

See Figure 7.3.4.

7.5.13		$$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$
$x>0$.
ⓘ Symbols: $F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E13 Encodings: TeX, pMML, png See also: Annotations for §7.5 and Ch.7

For $\mathrm{Ei}\left(x\right)$ see §6.2(i).

7.4 Symmetry7.6 Series Expansions

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