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		$\displaystyle\operatorname{erf}z$	$\displaystyle=\frac{1}{\sqrt{\pi}}\gamma\left(\tfrac{1}{2},z^{2}\right),$
ⓘ 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 $z$ : 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		$\displaystyle\operatorname{erfc}z$	$\displaystyle=\frac{1}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2},z^{2}\right),$
ⓘ 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 $z$ : 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		$\displaystyle\operatorname{erfc}z$	$\displaystyle=\frac{z}{\sqrt{\pi}}E_{\frac{1}{2}}\left(z^{2}\right).$
ⓘ 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 $z$ : 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 $z$ : 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 $z$ : 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		$C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2% }\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^% {2}\right).$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Permalink: http://dlmf.nist.gov/7.11.E6 Encodings: TeX, pMML, png See also: Annotations for §7.11, §7.11 and Ch.7

Generalized Hypergeometric Functions

ⓘ

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

For the notation see §§16.2(i) and 16.2(ii).

7.11.7		$\displaystyle C\left(z\right)$	$\displaystyle=z{{}_{1}F_{2}}\left(\tfrac{1}{4};\tfrac{5}{4},\tfrac{1}{2};-% \tfrac{1}{16}\pi^{2}z^{4}\right),$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/7.11.E7 Encodings: TeX, pMML, png See also: Annotations for §7.11, §7.11 and Ch.7
7.11.8		$\displaystyle S\left(z\right)$	$\displaystyle=\tfrac{1}{6}\pi z^{3}{{}_{1}F_{2}}\left(\tfrac{3}{4};\tfrac{7}{4% },\tfrac{3}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right).$
ⓘ Symbols: $S\left(\NVar{z}\right)$ : Fresnel integral, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable Permalink: http://dlmf.nist.gov/7.11.E8 Encodings: TeX, pMML, png See also: Annotations for §7.11, §7.11 and Ch.7

7.10 Derivatives 7.12 Asymptotic Expansions

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.8; Release date 2026-09-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov