5 Gamma Function Properties 5.9 Integral Representations 5.11 Asymptotic Expansions

§5.10 Continued Fractions

Notes:: See Wall (1948, pp. 364–365).
Keywords:: continued fractions, gamma function, psi function
Referenced by:: ¶ ‣ §3.10(ii), Other Changes
Permalink:: http://dlmf.nist.gov/5.10
Addition (effective with 1.0.5):: The reference to Mortici (2011) has been added at the end of this section.
Reported 2012-07-30

For $\realpart{z}>0$ ,

5.10.1 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z\right)+z-\left(z-% \tfrac{1}{2}\right)\mathop{\ln\/}\nolimits z-\tfrac{1}{2}\mathop{\ln\/}% \nolimits\!\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}{z+% \cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : complex variable and $a_{k}$ : coefficient
Permalink:: http://dlmf.nist.gov/5.10.E1
Encodings:: TeX, pMML, png

where

5.10.2

$a_{0}=\tfrac{1}{12},$

$a_{1}=\tfrac{1}{30},$

$a_{2}=\tfrac{53}{210},$

$a_{3}=\tfrac{195}{371},$

$a_{4}=\tfrac{22999}{22737},$

$a_{5}=\tfrac{299\;44523}{197\;33142},$

$a_{6}=\tfrac{10\;95352\;41009}{4\;82642\;75462}.$

Symbols:: $a_{k}$ : coefficient
A&S Ref:: 6.1.48
Permalink:: http://dlmf.nist.gov/5.10.E2
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

For exact values of $a_{7}$ to $a_{{11}}$ and 40S values of $a_{0}$ to $a_{{40}}$ , see Char (1980). Also see Cuyt et al. (2008, pp. 223–228), Jones and Thron (1980, pp. 348–350), Lorentzen and Waadeland (1992, pp. 221–224), and Mortici (2011) for further information.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 5.9 Integral Representations 5.11 Asymptotic Expansions