§7.2 Definitions

Referenced by:: §12.7(ii), §7.10
Permalink:: http://dlmf.nist.gov/7.2

§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

Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Keywords:: error functions
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

7.2.1 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2}{\sqrt{\pi}}\int _{0}^{z}e^{{-t^{2}}}dt,$

Defines:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.1.1
Referenced by:: §7.5, §7.6(ii)
Permalink:: http://dlmf.nist.gov/7.2.E1
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7.2.2 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{2}{\sqrt{\pi}}\int _{z}^{{\infty}}e^{{-t^{2}}}dt=1-\mathop{\mathrm{erf}\/}\nolimits z,$

Defines:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function
Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.1.2
Referenced by:: §7.5, §7.7(i)
Permalink:: http://dlmf.nist.gov/7.2.E2
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7.2.3 $\mathop{w\/}\nolimits\!\left(z\right)=e^{{-z^{2}}}\left(1+\frac{2i}{\sqrt{\pi}}\int _{0}^{z}e^{{t^{2}}}dt\right)=e^{{-z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-iz\right).$

Defines:: $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function
Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.1.3
Referenced by:: §7.19(i), §7.5
Permalink:: http://dlmf.nist.gov/7.2.E3
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$\mathop{\mathrm{erf}\/}\nolimits z$ , $\mathop{\mathrm{erfc}\/}\nolimits z$ , and $\mathop{w\/}\nolimits\!\left(z\right)$ are entire functions of $z$ , as is $\mathop{F\/}\nolimits\!\left(z\right)$ in the next subsection.

¶ Values at Infinity

Keywords:: error functions

7.2.4

$\lim _{{z\to\infty}}\mathop{\mathrm{erf}\/}\nolimits z=1,$

$\lim _{{z\to\infty}}\mathop{\mathrm{erfc}\/}\nolimits z=0$ , $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 7.1.16 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.2.E4
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§7.2(ii) Dawson’s Integral

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

7.2.5 $\mathop{F\/}\nolimits\!\left(z\right)=e^{{-z^{2}}}\int _{0}^{z}e^{{t^{2}}}dt.$

Defines:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.1.1
Permalink:: http://dlmf.nist.gov/7.2.E5
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§7.2(iii) Fresnel Integrals

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

7.2.6 $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)=\int _{z}^{\infty}e^{{\frac{1}{2}\pi it^{2}}}dt,$

Defines:: $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ : Fresnel integral
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
Referenced by:: §7.5, §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.2.E6
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7.2.7 $\mathop{C\/}\nolimits\!\left(z\right)=\int _{0}^{z}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi t^{2}\right)dt,$

Defines:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.3.1
Referenced by:: §7.5, §7.6(i)
Permalink:: http://dlmf.nist.gov/7.2.E7
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7.2.8 $\mathop{S\/}\nolimits\!\left(z\right)=\int _{0}^{z}\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi t^{2}\right)dt,$

Defines:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 7.3.2
Referenced by:: §7.5, §7.6(i)
Permalink:: http://dlmf.nist.gov/7.2.E8
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$\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ , $\mathop{C\/}\nolimits\!\left(z\right)$ , and $\mathop{S\/}\nolimits\!\left(z\right)$ are entire functions of $z$ , as are $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ in the next subsection.

¶ Values at Infinity

Keywords:: Fresnel integrals

7.2.9

$\lim _{{x\to\infty}}\mathop{C\/}\nolimits\!\left(x\right)=\tfrac{1}{2},$

$\lim _{{x\to\infty}}\mathop{S\/}\nolimits\!\left(x\right)=\tfrac{1}{2}.$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral and $x$ : real variable
A&S Ref:: 7.3.20
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.2.E9
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§7.2(iv) Auxiliary Functions

Keywords:: Fresnel integrals, auxiliary functions for Fresnel integrals
Permalink:: http://dlmf.nist.gov/7.2.iv

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

Defines:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals
Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 7.3.5
Referenced by:: §7.4, §7.5
Permalink:: http://dlmf.nist.gov/7.2.E10
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7.2.11 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\left(\tfrac{1}{2}-\mathop{C\/}\nolimits\!\left(z\right)\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-\mathop{S\/}\nolimits\!\left(z\right)\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right).$