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11: 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

… ►For these, the corresponding results for $U(a,z)$ with $a=2$, $\pm 3$, $-\frac{1}{2}$, $-\frac{3}{2}$, $-\frac{5}{2}$, and the corresponding results for $V(a,z)$ with $a=0$, $\pm 1$, $\pm 2$, $\pm 3$, $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, see Miller (1955, pp. 42–43 and 77–79). …

12: 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:

pMML, png, TeX

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:

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

►

Confluent Hypergeometric Functions

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13: 3.1 Arithmetics and Error Measures

§3.1 Arithmetics and Error Measures

… ►Symmetric rounding or rounding to nearest of $x$ gives $x_{-}$ or $x_{+}$, whichever is nearer to $x$, with maximum relative error equal to the machine precision $\frac{1}{2}{ϵ}_M=2^{-p}$. … ►

§3.1(v) Error Measures

… ►If $x\ne 0$, the relative error is … ►The mollified error is …

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

Encodings:

pMML, png, TeX

See also:

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:

pMML, png, TeX

See also:

Annotations for §7.4 and Ch.7

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$\mathrm{f}\left(\mathrm{i}z\right)=(1/\sqrt{2}){\mathrm{e}}^{\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}-\mathrm{i}\mathrm{f}\left(z\right),$

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$\mathrm{g}\left(\mathrm{i}z\right)=(1/\sqrt{2}){\mathrm{e}}^{-\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}+\mathrm{i}\mathrm{g}\left(z\right).$

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$\mathrm{f}\left(-z\right)=\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{f}\left(z\right),$

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15: 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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

pMML, png, TeX

See also:

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:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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16: 7.12 Asymptotic Expansions

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§7.12(i) Complementary Error Function

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). … ►(Note that some of these re-expansions themselves involve the complementary error function.) … ►as $z\to \mathrm{\infty }$ in . … ►where, for $n=0,1,2,\mathrm{\dots }$ and , …

17: 13.11 Series

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13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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$A_0=1,$

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$A_1=0,$

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$A_2={\displaystyle \frac{1}{2}}b,$

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$(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\mathrm{\dots }$.

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18: 7.20 Mathematical Applications

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§7.20(i) Asymptotics

►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). … ►Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. … ►

§7.20(iii) Statistics

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19: 13.6 Relations to Other Functions

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§13.6(ii) Incomplete Gamma Functions

… ►Special cases are the error functions ►

13.6.7 $$M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{\sqrt{\pi }}{2z}\mathrm{erf}\left(z\right),$$

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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 and $z$: complex variable

A&S Ref:

13.6.19

Referenced by:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.8 $$U(\frac{1}{2},\frac{1}{2},z^2)=\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}\left(z\right).$$

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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:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

… ►For the definition of ${}_2{}^{}F_0^{}(a,a-b+1;-;-z^{-1})$ when neither $a$ nor $a-b+1$ is a nonpositive integer see §16.5. …

20: 12.13 Sums

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12.13.2 $$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E2

Encodings:

pMML, png, TeX

See also:

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

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12.13.3 $$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E3

Encodings:

pMML, png, TeX

See also:

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

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12.13.5 $$U(a,x\cos t+y\sin t)={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),$$ $\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E5

Encodings:

pMML, png, TeX

See also:

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

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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). …