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

§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:: : differential of , : base of exponential function, $\int$ : 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

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:: : differential of , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function, $\int$ : 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

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, : differential of , : base of exponential function, $\int$ : integral and : complex variable
A&S Ref:: 7.1.3
Referenced by:: §7.19(i), §7.5
Permalink:: http://dlmf.nist.gov/7.2.E3
Encodings:: TeX, pMML, png

$\mathop{\mathrm{erf}\/}\nolimits z$ , $\mathop{\mathrm{erfc}\/}\nolimits z$ , and $\mathop{w\/}\nolimits\!\left(z\right)$ are entire functions of , 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 : 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

§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:: : differential of , : base of exponential function, $\int$ : integral and : complex variable
A&S Ref:: 7.1.1
Permalink:: http://dlmf.nist.gov/7.2.E5
Encodings:: TeX, pMML, png

§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:: : differential of , : base of exponential function, $\int$ : integral and : complex variable
Referenced by:: §7.5, §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.2.E6
Encodings:: TeX, pMML, png

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, : 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

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:: : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

$\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ , $\mathop{C\/}\nolimits\!\left(z\right)$ , and $\mathop{S\/}\nolimits\!\left(z\right)$ are entire functions of , 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 : 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

§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 : 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

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

Defines:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : 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 : complex variable
A&S Ref:: 7.3.6
Referenced by:: §7.4, §7.5
Permalink:: http://dlmf.nist.gov/7.2.E11
Encodings:: TeX, pMML, png

§7.2(v) Goodwin–Staton Integral

Keywords:: Goodwin–Staton integral
Permalink:: http://dlmf.nist.gov/7.2.v

7.2.12 $\mathop{G\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}\frac{e^{{-t^{2}}}}{t+z% }dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

Defines:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Goodwin–Staton integral
Symbols:: : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : complex variable
A&S Ref:: 27.6 (in different notation)
Permalink:: http://dlmf.nist.gov/7.2.E12
Encodings:: TeX, pMML, png

§7.2 Definitions

Contents

§7.2(i) Error Functions

¶ Values at Infinity

§7.2(ii) Dawson’s Integral

§7.2(iii) Fresnel Integrals

¶ Values at Infinity

§7.2(iv) Auxiliary Functions

§7.2(v) Goodwin–Staton Integral