§7.9 Continued Fractions

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)).
Keywords:: continued fractions, error functions
Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/7.9

7.9.1 $\sqrt{\pi}e^{{z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+\cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\realpart{z}>0$ ,

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\realpart{}$ : 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

7.9.2 $\sqrt{\pi}e^{{z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\realpart{z}>0$ ,

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §7.9
Permalink:: http://dlmf.nist.gov/7.9.E2
Encodings:: TeX, pMML, png

7.9.3 $\mathop{w\/}\nolimits\!\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1}{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\imagpart{z}>0$ .

Symbols:: $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, $\imagpart{}$ : imaginary part 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 Cuyt et al. (2008, pp. 255–260, 263–267, 270–273).