7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.1 Special Notation 7.3 Graphics

§7.2 Definitions

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Referenced by:: §12.7(ii), §7.10
Permalink:: http://dlmf.nist.gov/7.2
See also:: Annotations for Ch.7

§7.2(i) Error Functions
§7.2(ii) Dawson’s Integral
§7.2(iii) Fresnel Integrals
§7.2(iv) Auxiliary Functions
§7.2(v) Goodwin–Staton Integral

§7.2(i) Error Functions

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Keywords:: Faddeeva (or Faddeyeva) function, definition, definitions, error functions
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §12.13(i), §13.18(ii), §13.6(ii), §2.11(iii), §32.10(iv), §8.18(ii), §8.4
Permalink:: http://dlmf.nist.gov/7.2.i
See also:: Annotations for §7.2 and Ch.7

7.2.1 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,\mathrm{d}t,$

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Defines:: $\operatorname{erf}\NVar{z}$ : error function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : complex variable
A&S Ref:: 7.1.1
Referenced by:: §7.5, §7.6(ii)
Permalink:: http://dlmf.nist.gov/7.2.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(i), §7.2 and Ch.7

7.2.2 $\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{% d}t=1-\operatorname{erf}z,$

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Defines:: $\operatorname{erfc}\NVar{z}$ : complementary error function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : complex variable
A&S Ref:: 7.1.2
Referenced by:: §7.5, §7.7(i)
Permalink:: http://dlmf.nist.gov/7.2.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(i), §7.2 and Ch.7

7.2.3 $w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\,% \mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right).$

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Defines:: $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : complex variable
Keywords:: Faddeeva (or Faddeyeva) function, definition
A&S Ref:: 7.1.3
Referenced by:: §7.19(i), §7.5, Erratum (V1.0.17) for Equation (7.2.3)
Permalink:: http://dlmf.nist.gov/7.2.E3
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.17):: Originally named as a complementary error function, this has been renamed as the Faddeeva (or Faddeyeva) function.
See also:: Annotations for §7.2(i), §7.2 and Ch.7

$\operatorname{erf}z$ , $\operatorname{erfc}z$ , and $w\left(z\right)$ are entire functions of , as is $F\left(z\right)$ in the next subsection.

Values at Infinity

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Keywords:: error functions, values at infinity
See also:: Annotations for §7.2(i), §7.2 and Ch.7

7.2.4

$\lim_{z\to\infty}\operatorname{erf}z=1,$

$\lim_{z\to\infty}\operatorname{erfc}z=0$ , $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\operatorname{erf}\NVar{z}$ : error function, $\operatorname{ph}$ : phase and : complex variable
A&S Ref:: 7.1.16 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.2.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §7.2(i), §7.2(i), §7.2 and Ch.7

§7.2(ii) Dawson’s Integral

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Keywords:: Dawson’s integral, definition
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §8.11(iv), §8.12, §8.4
Permalink:: http://dlmf.nist.gov/7.2.ii
See also:: Annotations for §7.2 and Ch.7

7.2.5 $F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\,\mathrm{d}t.$

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Defines:: $F\left(\NVar{z}\right)$ : Dawson’s integral
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : complex variable
A&S Ref:: 7.1.1
Permalink:: http://dlmf.nist.gov/7.2.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(ii), §7.2 and Ch.7

§7.2(iii) Fresnel Integrals

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Keywords:: Fresnel integrals, definition
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §11.10(vi)
Permalink:: http://dlmf.nist.gov/7.2.iii
See also:: Annotations for §7.2 and Ch.7

7.2.6 $\mathcal{F}\left(z\right)=\int_{z}^{\infty}e^{\tfrac{1}{2}\pi\mathrm{i}t^{2}}% \,\mathrm{d}t,$

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Defines:: $\mathcal{F}\left(\NVar{z}\right)$ : Fresnel integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : complex variable
Referenced by:: §7.5, §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.2.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(iii), §7.2 and Ch.7

7.2.7 $C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2}\pi t^{2}\right)\,\mathrm{d}t,$

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Defines:: $C\left(\NVar{z}\right)$ : Fresnel integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : complex variable
A&S Ref:: 7.3.1
Referenced by:: §7.5, §7.6(i)
Permalink:: http://dlmf.nist.gov/7.2.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(iii), §7.2 and Ch.7

7.2.8 $S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\,\mathrm{d}t,$

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Defines:: $S\left(\NVar{z}\right)$ : Fresnel integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function and : complex variable
A&S Ref:: 7.3.2
Referenced by:: §7.5, §7.6(i)
Permalink:: http://dlmf.nist.gov/7.2.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(iii), §7.2 and Ch.7

$\mathcal{F}\left(z\right)$ , $C\left(z\right)$ , and $S\left(z\right)$ are entire functions of , as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection.

Values at Infinity

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Keywords:: Fresnel integrals, values at infinity
See also:: Annotations for §7.2(iii), §7.2 and Ch.7

7.2.9

$\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2},$

$\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}.$

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Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral and : real variable
A&S Ref:: 7.3.20
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.2.E9
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §7.2(iii), §7.2(iii), §7.2 and Ch.7

§7.2(iv) Auxiliary Functions

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Keywords:: Fresnel integrals, auxiliary functions, auxiliary functions for Fresnel integrals, definitions, relations to other functions
Permalink:: http://dlmf.nist.gov/7.2.iv
See also:: Annotations for §7.2 and Ch.7

7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$

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Defines:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals
Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and : complex variable
A&S Ref:: 7.3.5
Referenced by:: §7.4, §7.5
Permalink:: http://dlmf.nist.gov/7.2.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §7.2(iv), §7.2 and Ch.7

7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$

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