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 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$ ,

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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 : 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 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$ ,

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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 : 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 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$ .

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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 : 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).

7.8 Inequalities 7.10 Derivatives

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