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11: 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

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See also:

Annotations for §7.10 and Ch.7

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7.10.2 $$w^{\prime }\left(z\right)=-2zw\left(z\right)+(2\mathrm{i}/\sqrt{\pi }),$$

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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, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E2

Encodings:

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Annotations for §7.10 and Ch.7

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7.10.3 $$w^{(n+2)}\left(z\right)+2z{w}^{(n+1)}\left(z\right)+2(n+1)w^{(n)}\left(z\right)=0,$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$w\left(z\right)$: Faddeeva (or Faddeyeva) function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E3

Encodings:

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See also:

Annotations for §7.10 and Ch.7

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12: 7.17 Inverse Error Functions

§7.17 Inverse Error Functions

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§7.17(i) Notation

►The inverses of the functions $x=\mathrm{erf}y$, $x=\mathrm{erfc}y$, $y\in ℝ$, are denoted by … ►

§7.17(ii) Power Series

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§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

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13: 7.3 Graphics

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Figure 7.3.1: Complementary error functions $\mathrm{erfc}x$ and $\mathrm{erfc}\left(10x\right)$, $-3\le x\le 3$. Magnify

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Figure 7.3.5: $|\mathrm{erf}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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Figure 7.3.6: $|\mathrm{erfc}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

14: 7.23 Tables

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Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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Abramowitz and Stegun (1964, Chapter 7) includes $w\left(z\right)$, $x=0(.1)3.9$, $y=0(.1)3$, 6D.

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Fettis et al. (1973) gives the first 100 zeros of $\mathrm{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\mathrm{erfc}z$), 11S.

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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\mathrm{erf}z$, 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$, 8S.

15: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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16: 7.25 Software

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§7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$

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§7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$

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17: 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

Encodings:

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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

Encodings:

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See also:

Annotations for §7.9 and Ch.7

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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). … ►

§7.20(iii) Statistics

… ►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

19: 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:

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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:

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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

Encodings:

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See also:

Annotations for §7.4 and Ch.7

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20: 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:

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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:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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