Digital Library of Mathematical Functions
The Gamma Function -- Continued Fractions

### Continued Fractions

For ℜ z > 0 ,

ln Γ ( z ) + z - ( z - 1 2 ) ln z - 1 2 ln ( 2 π ) = a 0 + z + a 1 + z + a 2 + z + a 3 + z + a 4 + z + a 5 z + ⋯
a 0 = 1 12 , a 1 = 1 30 , a 2 = 53 210 , a 3 = 195 371 , a 4 = 22999 22737 , a 5 = 299 44523 197 33142 , a 6 = 10 95352 41009 4 82642 75462

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.