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

¶ Incomplete Gamma Functions and Generalized Exponential Integral

Keywords:: error functions, generalized exponential integral, incomplete gamma functions

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

7.11.1 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{1}{\sqrt{\pi}}\mathop{\gamma\/}% \nolimits\!\left(\tfrac{1}{2},z^{2}\right),$

Symbols:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E1
Encodings:: TeX, pMML, png

7.11.2 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{1}{\sqrt{\pi}}\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2},z^{2}\right),$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
Referenced by:: §7.12(i)
Permalink:: http://dlmf.nist.gov/7.11.E2
Encodings:: TeX, pMML, png

7.11.3 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{z}{\sqrt{\pi}}\mathop{E_{{\frac{1}{2% }}}\/}\nolimits\!\left(z^{2}\right).$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral and : complex variable
Referenced by:: §7.12(i)
Permalink:: http://dlmf.nist.gov/7.11.E3
Encodings:: TeX, pMML, png

¶ Confluent Hypergeometric Functions

Keywords:: Fresnel integrals, confluent hypergeometric functions, error functions

For the notation see §13.2(i).

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function and : complex variable
A&S Ref:: 7.1.21
Permalink:: http://dlmf.nist.gov/7.11.E4
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : base of exponential function and : complex variable
Referenced by:: §7.11
Permalink:: http://dlmf.nist.gov/7.11.E5
Encodings:: TeX, pMML, png

7.11.6 $\mathop{C\/}\nolimits\!\left(z\right)+i\mathop{S\/}\nolimits\!\left(z\right)=z% \mathop{M\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}% \right)=ze^{{\pi iz^{2}/2}}\mathop{M\/}\nolimits\!\left(1,\tfrac{3}{2},-\tfrac% {1}{2}\pi iz^{2}\right).$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function and : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E6
Encodings:: TeX, pMML, png

¶ Generalized Hypergeometric Functions

Keywords:: Fresnel integrals, generalized hypergeometric functions

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

7.11.7 $\mathop{C\/}\nolimits\!\left(z\right)=z\mathop{{{}_{{1}}F_{{2}}}\/}\nolimits\!% \left(\tfrac{1}{4};\tfrac{5}{4},\tfrac{1}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right),$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E7
Encodings:: TeX, pMML, png

7.11.8 $\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{6}\pi z^{3}\mathop{{{}_{{1}}F_% {{2}}}\/}\nolimits\!\left(\tfrac{3}{4};\tfrac{7}{4},\tfrac{3}{2};-\tfrac{1}{16% }\pi^{2}z^{4}\right).$

Symbols:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E8
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 7.10 Derivatives 7.12 Asymptotic Expansions