§8.9 Continued Fractions

Keywords:: incomplete gamma functions
Referenced by:: ¶ ‣ §3.10(ii), ¶ ‣ §3.10(iii)
Permalink:: http://dlmf.nist.gov/8.9

8.9.1 $\mathop{\Gamma\/}\nolimits\!\left(a+1\right)e^{z}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=\cfrac{1}{1-\cfrac{z}{a+1+\cfrac{z}{a+2-\cfrac{(a+1)z}{a+3+\cfrac{2z}{a+4-\cfrac{(a+2)z}{a+5+\cfrac{3z}{a+6-\cdots}}}}}}},$ $a\neq-1,-2,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : 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

8.9.2 $z^{{-a}}e^{z}\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\cfrac{z^{{-1}}}{1+\cfrac{(1-a)z^{{-1}}}{1+\cfrac{z^{{-1}}}{1+\cfrac{(2-a)z^{{-1}}}{1+\cfrac{2z^{{-1}}}{1+\cfrac{(3-a)z^{{-1}}}{1+\cfrac{3z^{{-1}}}{1+\cdots}}}}}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.19(vii)
Permalink:: http://dlmf.nist.gov/8.9.E2
Encodings:: TeX, pMML, png

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