§ 5.10. Continued Fractions

Notes:: See Wall (1948, pp. 364–365).
Keywords:: gamma function
Permalink:: http://dlmf.nist.gov/5.10

For $\realpart{z}>0$ ,

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

Defines:: $a_{k}$ : coefficient
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.10.E1
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where

5.10.2

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

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

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

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

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

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

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

Defines:: $a_{k}$ : coefficient
A&S Ref:: 6.1.48
Permalink:: http://dlmf.nist.gov/5.10.E2
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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), and Lorentzen and Waadeland (1992, pp. 221–224) for further information.