complementary error function

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11: 7.2 Definitions

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7.2.2 $$\mathrm{erfc}z=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=1-\mathrm{erf}z,$$

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Defines:

$\mathrm{erfc}z$: complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.2

Referenced by:

§7.5, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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7.2.3 $$w\left(z\right)={\mathrm{e}}^{-z^2}\left(1+\frac{2\mathrm{i}}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{t^2}dt\right)={\mathrm{e}}^{-z^2}\mathrm{erfc}\left(-\mathrm{i}z\right).$$

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Defines:

$w\left(z\right)$: Faddeeva (or Faddeyeva) function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

definition, Faddeeva (or Faddeyeva) function

A&S Ref:

7.1.3

Referenced by:

§7.19(i), §7.5, Erratum (V1.0.17) for Equation (7.2.3)

Permalink:

http://dlmf.nist.gov/7.2.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally named as a complementaryerrorfunction, this has been renamed as the Faddeeva (or Faddeyeva) function.

See also:

Annotations for §7.2(i), §7.2 and Ch.7

► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. … ►

$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z=0$ , .

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12: 7.4 Symmetry

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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:

pMML, png, TeX

See also:

Annotations for §7.4 and Ch.7

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13: 12.7 Relations to Other Functions

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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12.7.5 $$U(\frac{1}{2},z)=D_{-1}\left(z\right)=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{\frac{1}{4}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right),$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder 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, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Referenced by:

§12.11(ii)

Permalink:

http://dlmf.nist.gov/12.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.6 $$U(n+\frac{1}{2},z)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi }{2}}\frac{{(-1)}^n}{n!}{\mathrm{e}}^{-\frac{1}{4}{z}^2}\frac{d^n\left({\mathrm{e}}^{\frac{1}{2}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right)\right)}{{dz}^n},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.7 $$U(n+\frac{1}{2},z)={\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{𝐻ℎ}}_n\left(z\right)=\sqrt{\pi }{\mathrm{\hspace{0.17em}2}}^{\frac{1}{2}(n-1)}{\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{i}}^n\mathrm{erfc}\left(z/\sqrt{2}\right),$$ $n=-1,0,1,\mathrm{\dots }$.

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Symbols:

${\mathrm{𝐻ℎ}}_n\left(z\right)$: probability function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/12.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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14: 7.11 Relations to Other Functions

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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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.5 $$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(\frac{1}{2},\frac{1}{2},z^2)=\frac{z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(1,\frac{3}{2},z^2).$$

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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:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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15: 7.14 Integrals

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7.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}at}\mathrm{erfc}\left(bt\right)dt=\frac{1}{a\sqrt{\pi }}F\left(\frac{a}{b}\right)+\frac{\mathrm{i}}{2a}\left(1-{\mathrm{e}}^{-{(a/b)}^2}\right),$$ $a\in ℂ$, .

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.18 (in different form)

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E1

Encodings:

pMML, png, TeX

See also:

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

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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:

pMML, png, TeX

See also:

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

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7.14.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{(a-b)t}\mathrm{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)dt=\frac{{\mathrm{e}}^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})},$$ , $\mathrm{\Re }b>0$, $\mathrm{\Re }c\ge 0$.

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.21

Referenced by:

§3.5(ix), §7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E4

Encodings:

pMML, png, TeX

See also:

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

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16: 12.13 Sums

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12.13.6 $$n!U(n+\frac{1}{2},z)={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{2}{z}^2}\mathrm{erfc}(z/\sqrt{2})U(-n-\frac{1}{2},\mathrm{i}z)+\sum _{m=1}^{\lfloor \frac{1}{2}n+\frac{1}{2}\rfloor }U(2m-n-\frac{1}{2},z),$$ $n=0,1,2,\mathrm{\dots }.$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►For $\mathrm{erfc}$ see §7.2(i). …

17: 7.7 Integral Representations

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7.7.1 $$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

7.4.11 (in different form)

Referenced by:

§7.11, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.4 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{\sqrt{t+z^2}}dt=\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{a{z}^2}\mathrm{erfc}\left(\sqrt{a}z\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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Symbols:

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

Keywords:

Laplace transform

A&S Ref:

7.4.8

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.6 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-(a{t}^2+2bt+c)}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{(b^2-ac)/a}\mathrm{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

A&S Ref:

7.4.32 (in different form)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.7 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{4a}\left({\mathrm{e}}^{2ab}\mathrm{erfc}\left(ax+(b/x)\right)+{\mathrm{e}}^{-2ab}\mathrm{erfc}\left(ax-(b/x)\right)\right),$$ $x>0$, .

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $x$: real variable

A&S Ref:

7.4.33 (in different form)

Referenced by:

§7.7(i), §7.8

Permalink:

http://dlmf.nist.gov/7.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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18: 7.13 Zeros

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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 $\mathrm{erfc}z.$

► ►►

$n$	$x_n$	$y_n$
…

►

… ►In consequence of (7.5.5) and (7.5.10), zeros of $ℱ\left(z\right)$ are related to zeros of $\mathrm{erfc}z$. Thus if $z_n$ is a zero of $\mathrm{erfc}z$ (§7.13(ii)), then $(1+\mathrm{i})z_n/\sqrt{\pi }$ is a zero of $ℱ\left(z\right)$. …

19: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.3 $$P(a,z)=\frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{a/2}\right)-S(a,\eta ),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.4 $$Q(a,z)=\frac{1}{2}\mathrm{erfc}\left(\eta \sqrt{a/2}\right)+S(a,\eta ),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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$d(\pm \chi )=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{{\chi }^2/2}\mathrm{erfc}\left(\pm \chi /\sqrt{2}\right),$

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). … ►These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

20: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►

7.20.1 $$\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{\mathrm{e}}^{-{(t-m)}^2/(2{\sigma }^2)}dt=\frac{1}{2}\mathrm{erfc}\left(\frac{m-x}{\sigma \sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma }\right)=P\left(\frac{x-m}{\sigma }\right).$$

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable, $m$: mean and $\sigma$: standard deviation

A&S Ref:

7.1.22 (slightly modified)

Permalink:

http://dlmf.nist.gov/7.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.20(iii), §7.20 and Ch.7

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