7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.10 Derivatives 7.12 Asymptotic Expansions

§7.11 Relations to Other Functions

ⓘ

Notes:: These results may be verified by comparing the power-series expansions of both sides of each equation. For (7.11.5) use (7.7.1) and (13.4.4).
Permalink:: http://dlmf.nist.gov/7.11
See also:: Annotations for Ch.7

Incomplete Gamma Functions and Generalized Exponential Integral

ⓘ

Keywords:: error functions, generalized exponential integral, generalized exponential integrals, incomplete gamma functions, relations to other functions
See also:: Annotations for §7.11 and Ch.7

For the notation see §§8.2(i) and 8.19(i).

7.11.1 $\operatorname{erf}z=\frac{1}{\sqrt{\pi}}\gamma\left(\tfrac{1}{2},z^{2}\right),$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §7.11, §7.11 and Ch.7

7.11.2 $\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2},z^{2}\right),$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and : complex variable
Referenced by:: §7.12(i)
Permalink:: http://dlmf.nist.gov/7.11.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §7.11, §7.11 and Ch.7

7.11.3 $\operatorname{erfc}z=\frac{z}{\sqrt{\pi}}E_{\frac{1}{2}}\left(z^{2}\right).$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral and : complex variable
Referenced by:: §7.12(i)
Permalink:: http://dlmf.nist.gov/7.11.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §7.11, §7.11 and Ch.7

Confluent Hypergeometric Functions

ⓘ

Keywords:: Fresnel integrals, confluent hypergeometric functions, error functions, relations to other functions
See also:: Annotations for §7.11 and Ch.7

For the notation see §13.2(i).

7.11.4 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{% 2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}\right),$

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and : complex variable
A&S Ref:: 7.1.21
Permalink:: http://dlmf.nist.gov/7.11.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §7.11, §7.11 and Ch.7

7.11.5 $\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},\tfrac{% 1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},z^{2}% \right).$

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm and : complex variable
Referenced by:: §7.11
Permalink:: http://dlmf.nist.gov/7.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §7.11, §7.11 and Ch.7

7.11.6 $C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2% }\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^% {2}\right).$

ⓘ