7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.10 Derivatives7.12 Asymptotic Expansions

§7.11 Relations to Other Functions

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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

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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		$\mathrm{erf}z$	$={\displaystyle \frac{1}{\sqrt{\pi }}}\gamma (\frac{1}{2},z^2),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: 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		$\mathrm{erfc}z$	$={\displaystyle \frac{1}{\sqrt{\pi }}}\mathrm{\Gamma }(\frac{1}{2},z^2),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{\Gamma }(a,z)$: 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		$\mathrm{erfc}z$	$={\displaystyle \frac{z}{\sqrt{\pi }}}{E}_{\frac{1}{2}}\left(z^2\right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $E_p\left(z\right)$: 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

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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		$$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}M(1,\frac{3}{2},z^2),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: 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		$$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(\frac{1}{2},\frac{1}{2},z^2)=\frac{z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(1,\frac{3}{2},z^2).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: 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		$$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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

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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		$C\left(z\right)$	$=z{}_1{}^{}F_2^{}(\frac{1}{4};\frac{5}{4},\frac{1}{2};-\frac{1}{16}{\pi }^2{z}^4),$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ 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		$S\left(z\right)$	$=\frac{1}{6}\pi z^3{}_1{}^{}F_2^{}(\frac{3}{4};\frac{7}{4},\frac{3}{2};-\frac{1}{16}{\pi }^2{z}^4).$
ⓘ Symbols: $S\left(z\right)$: Fresnel integral, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ 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 Derivatives7.12 Asymptotic Expansions

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