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5 Gamma FunctionProperties

§5.10 Continued Fractions

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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,
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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
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}.
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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).

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