derivatives of the error function

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1: 7.10 Derivatives

§7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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2: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iii) Properties

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7.18.3 $$\frac{d}{dz}{\mathrm{i}}^n\mathrm{erfc}\left(z\right)=-{\mathrm{i}}^{n-1}\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.8

Permalink:

http://dlmf.nist.gov/7.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.4 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{z^2}\mathrm{erfc}z\right)={(-1)}^n{2}^{n}n!{\mathrm{e}}^{z^2}{\mathrm{i}}^n\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.9

Permalink:

http://dlmf.nist.gov/7.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.5 $$\frac{d^{2}W}{{dz}^2}+2z\frac{dW}{dz}-2nW=0,$$ $W(z)=A{\mathrm{i}}^n\mathrm{erfc}\left(z\right)+B{\mathrm{i}}^n\mathrm{erfc}\left(-z\right)$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.2

Permalink:

http://dlmf.nist.gov/7.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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3: 12.7 Relations to Other Functions

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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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4: 10.21 Zeros

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5: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). … ►

6: 7.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. …

7: 10.40 Asymptotic Expansions for Large Argument

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$\nu$-Derivative

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§10.40(ii) Error Bounds for Real Argument and Order

… ►For the error term in (10.40.1) see §10.40(iii). ►

§10.40(iii) Error Bounds for Complex Argument and Order

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8: 2.7 Differential Equations

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2.7.24 $$F(x)=\int \left(\frac{1}{f^{1/4}}\frac{d^2}{{dx}^2}\left(\frac{1}{f^{1/4}}\right)-\frac{g}{f^{1/2}}\right)dx,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral and $F$: error-control function

Permalink:

http://dlmf.nist.gov/2.7.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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9: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

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§13.3(i) Recurrence Relations

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§13.3(ii) Differentiation Formulas

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13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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13.3.29 $${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.15(ii), §13.3(ii)

Permalink:

http://dlmf.nist.gov/13.3.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

10: 10.63 Recurrence Relations and Derivatives

§10.63 Recurrence Relations and Derivatives

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§10.63(i) ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$

►Let $f_{\nu }(x)$, $g_{\nu }(x)$ denote any one of the ordered pairs: … ►

§10.63(ii) Cross-Products

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