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7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.8 Inequalities 7.10 Derivatives

§7.9 Continued Fractions

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Keywords:: continued fractions, error functions
Notes:: See Nielsen (1906a, p. 217) and Lorentzen and Waadeland (1992, pp. 576–577). (7.9.2) is the even part of (7.9.1) (compare §1.12(iv)).
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/7.9
See also:: Annotations for Ch.7

7.9.1		$\displaystyle\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z$	$\displaystyle=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+\cfrac{1}{z^{2}+\cfrac{% \frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$
$\Re z>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $\Re$ : 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: TeX, pMML, png See also: Annotations for §7.9 and Ch.7
7.9.2		$\displaystyle\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z$	$\displaystyle=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2z^{2}+5-\cfrac{3\cdot 4}{2% z^{2}+9-\cdots}}},$
$\Re z>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $\Re$ : real part and $z$ : complex variable Referenced by: §7.9 Permalink: http://dlmf.nist.gov/7.9.E2 Encodings: TeX, pMML, png See also: Annotations for §7.9 and Ch.7
7.9.3		$\displaystyle w\left(z\right)$	$\displaystyle=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1}{% z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$
$\Im z>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\Im$ : 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: TeX, pMML, png See also: Annotations for §7.9 and Ch.7

See also Cuyt et al. (2008, pp. 255–260, 263–267, 270–273).

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