DLMF: GA.10 Continued Fractions

§GA.10. Continued Fractions

Sources:

Wall(1948)(Chapter 19).

Notes:

Wall(1948)(pp. 364–365)

Keywords:

continued fraction, gamma function

For $\realpart{z}>0$ ,

GA.10.1 $\mathrm{ln}\Gamma\!\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\mathrm{ln}z-\tfrac{1}{2}\mathrm{ln}\!\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}{z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable and $a_{k}$ : coefficient
Encodings:: pMathML, png, TeX

GA.10.2 $a_{0}=\tfrac{1}{12},$ $a_{1}=\tfrac{1}{30},$ $a_{2}=\tfrac{53}{210},$ $a_{3}=\tfrac{195}{371},$ $a_{4}=\tfrac{22999}{22737},$ $a_{5}=\tfrac{299\; 44523}{197\; 33142},$ $a_{6}=\tfrac{10\; 95352\; 41009}{4\; 82642\; 75462}.$

Notations:: $a_{k}$ : coefficient
A&S Ref:: 6.1.48
Encodings:: pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, pMathML, png, png, png, png, png, png, png, TeX, TeX, TeX, TeX, TeX, TeX, TeX

For rational values of $a_{7}$ to $a_{{11}}$ and 40S values of $a_{0}$ to $a_{{40}}$ , see Char(1980). Also see Jones and Thron(1980)(pp. 348–350) and Lorentzen and Waadeland(1992)(pp. 221–224) for further information.