5 Gamma Function Properties 5.9 Integral Representations 5.11 Asymptotic Expansions

§5.10 Continued Fractions

ⓘ

Keywords:: continued fraction, continued fractions, gamma function, logarithm, psi function
Notes:: See Wall (1948, pp. 364–365).
Referenced by:: §3.10(ii), Erratum (V1.0.10) for References, Erratum (V1.0.17) for References, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/5.10
Clarification (effective with 1.0.17):: A mention of the references Mortici (2011a, 2013a) was removed and the reference Jones and Van Assche (1998) was added.
Addition (effective with 1.0.10):: The reference to Mortici (2013a) was added at the end of this subsection.
Addition (effective with 1.0.5):: The reference to Mortici (2011a) has been added at the end of this section.
See also:: Annotations for Ch.5

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: Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8) 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. Suggested 2015-02-13 by Philippe Spindel See also: Annotations for §5.10 and Ch.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 Ch.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 Jones and Van Assche (1998).

5.9 Integral Representations 5.11 Asymptotic Expansions

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