error functions

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11: 7.22 Methods of Computation

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§7.22(i) Main Functions

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§7.22(iii) Repeated Integrals of the Complementary Error Function

►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. … ►The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …

12: 7.9 Continued Fractions

§7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

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Annotations for §7.9 and Ch.7

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

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See also:

Annotations for §7.9 and Ch.7

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7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

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See also:

Annotations for §7.9 and Ch.7

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13: 7.15 Sums

§7.15 Sums

►For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).

14: 7.4 Symmetry

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7.4.1 $$\mathrm{erf}\left(-z\right)=-\mathrm{erf}\left(z\right),$$

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Symbols:

$\mathrm{erf}z$: error function and $z$: complex variable

A&S Ref:

7.1.9

Permalink:

http://dlmf.nist.gov/7.4.E1

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Annotations for §7.4 and Ch.7

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7.4.2 $$\mathrm{erfc}\left(-z\right)=2-\mathrm{erfc}\left(z\right),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.4.E2

Encodings:

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See also:

Annotations for §7.4 and Ch.7

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7.4.3 $$w\left(-z\right)=2{\mathrm{e}}^{-z^2}-w\left(z\right).$$

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Symbols:

$w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

7.1.11

Permalink:

http://dlmf.nist.gov/7.4.E3

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See also:

Annotations for §7.4 and Ch.7

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15: 7.16 Generalized Error Functions

§7.16 Generalized Error Functions

►Generalizations of the error function and Dawson’s integral are ${\int }_0^x{\mathrm{e}}^{-t^p}dt$ and ${\int }_0^x{\mathrm{e}}^{t^p}dt$. …

16: 7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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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:

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See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.2 $$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}\mathrm{\Gamma }(\frac{1}{2},z^2),$$

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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:

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See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.3 $$\mathrm{erfc}z=\frac{z}{\sqrt{\pi }}{E}_{\frac{1}{2}}\left(z^2\right).$$

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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:

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See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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17: 7.5 Interrelations

§7.5 Interrelations

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7.5.1 $$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E1

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Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

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Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

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Annotations for §7.5 and Ch.7

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18: 7.6 Series Expansions

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§7.6(i) Power Series

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7.6.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{n!(2n+1)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.5

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E1

Encodings:

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See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.2 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}\sum _{n=0}^{\mathrm{\infty }}\frac{2^n{z}^{2n+1}}{1\cdot 3\mathrm{\cdots }(2n+1)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.6

Referenced by:

§7.6(i), §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E2

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Annotations for §7.6(i), §7.6 and Ch.7

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7.6.3 $$w\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(\mathrm{i}z)}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.8

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E3

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Annotations for §7.6(i), §7.6 and Ch.7

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§7.6(ii) Expansions in Series of Spherical Bessel Functions

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19: 7.13 Zeros

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§7.13(i) Zeros of $\mathrm{erf}z$

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Table 7.13.1: Zeros $x_n+\mathrm{i}{y}_n$ of $\mathrm{erf}z.$

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$n$	$x_n$	$y_n$
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§7.13(ii) Zeros of $\mathrm{erfc}z$

… ►The other zeros of $\mathrm{erfc}z$ are ${\overline{z}}_n$. … ►

Table 7.13.2: Zeros $x_n+\mathrm{i}{y}_n$ of $\mathrm{erfc}z.$

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$n$	$x_n$	$y_n$
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20: 7.14 Integrals

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§7.14(i) Error Functions

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Fourier Transform

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Laplace Transforms

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7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

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See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.19

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E3

Encodings:

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See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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