# §5.10 Continued Fractions

For $\Re z>0$,

 5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{Ln}\NVar{z}$: general logarithm function, $\ln\NVar{z}$: principal branch of logarithm function, $z$: complex variable and $a_{k}$: coefficient Referenced by: Other Changes Permalink: http://dlmf.nist.gov/5.10.E1 Encodings: TeX, pMML, png Addition (effective with 1.0.10): To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$, where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm. Reported 2015-02-13 by Philippe Spindel See also: Annotations for 5.10 and 5

where

 5.10.2 $\displaystyle a_{0}$ $\displaystyle=\tfrac{1}{12},$ $\displaystyle a_{1}$ $\displaystyle=\tfrac{1}{30},$ $\displaystyle a_{2}$ $\displaystyle=\tfrac{53}{210},$ $\displaystyle a_{3}$ $\displaystyle=\tfrac{195}{371},$ $\displaystyle a_{4}$ $\displaystyle=\tfrac{22999}{22737},$ $\displaystyle a_{5}$ $\displaystyle=\tfrac{299\;44523}{197\;33142},$ $\displaystyle a_{6}$ $\displaystyle=\tfrac{10\;95352\;41009}{4\;82642\;75462}.$ ⓘ 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 See also: Annotations for 5.10 and 5

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 (2011a, 2013a).