§8.9 Continued Fractions

ⓘ

8.9.1	$\displaystyle\Gamma\left(a+1\right)e^{z}\gamma^{*}\left(a,z\right)$	$\displaystyle=\cfrac{1}{1-\cfrac{z}{a+1+\cfrac{z}{a+2-\cfrac{(a+1)z}{a+3+% \cfrac{2z}{a+4-\cfrac{(a+2)z}{a+5+\cfrac{3z}{a+6-\cdots}}}}}}},$
$a\neq-1,-2,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter A&S Ref: 6.5.31 (in modified form.) Permalink: http://dlmf.nist.gov/8.9.E1 Encodings: TeX, pMML, png See also: Annotations for §8.9 and Ch.8
8.9.2	$\displaystyle z^{-a}e^{z}\Gamma\left(a,z\right)$	$\displaystyle=\cfrac{z^{-1}}{1+\cfrac{(1-a)z^{-1}}{1+\cfrac{z^{-1}}{1+\cfrac{(% 2-a)z^{-1}}{1+\cfrac{2z^{-1}}{1+\cfrac{(3-a)z^{-1}}{1+\cfrac{3z^{-1}}{1+\cdots% }}}}}}},$
$\|\operatorname{ph}z\|<\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : parameter Referenced by: §8.19(vii) Permalink: http://dlmf.nist.gov/8.9.E2 Encodings: TeX, pMML, png See also: Annotations for §8.9 and Ch.8

For these expansions and further information see Jones and Thron (1985). See also Cuyt et al. (2008, pp. 240–251).