§5.9 Integral Representations§5.11 Asymptotic Expansions

§ 5.10. Continued Fractions

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,

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}.

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.