Faddeeva (or Faddeyeva) function

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1: 7.21 Physical Applications

§7.21 Physical Applications

… ►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). …Efficient algorithms for computing the Faddeeva (or Faddeyeva) function are discussed in Wells (1999), a paper frequently cited in the astrophysics literature. …

2: 7.10 Derivatives

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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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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3: 7.2 Definitions

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§7.2(i) Error Functions

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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:

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Clarification (effective with 1.0.17):

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

See also:

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

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4: 7.9 Continued Fractions

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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:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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5: Bibliography Z

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Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

2nd item, 5th item.

Encodings:

BibTeX

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Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry

Referenced by:

3rd item, 6th item, §7.25(iii), §7.25(vi).

Encodings:

BibTeX

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Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

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External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §7.25(iii).

Encodings:

BibTeX

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6: 7.4 Symmetry

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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:

pMML, png, TeX

See also:

Annotations for §7.4 and Ch.7

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7: 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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7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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8: Bibliography F

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Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

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Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

Encodings:

BibTeX

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

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9: 7.6 Series Expansions

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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

Encodings:

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See also:

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

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10: 7.7 Integral Representations

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7.7.2 $$w\left(z\right)=\frac{1}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t-z}=\frac{2z}{\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t^2-z^2},$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.4 (in different form)

Referenced by:

§7.19(i), §7.22(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E2

Encodings:

pMML, png, TeX

See also:

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

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