Digital Library of Mathematical Functions
The Gamma Function -- Approximations

### Approximations

###### Contents

- Rational Approximations
- Expansions in Chebyshev Series
- Approximations in the Complex Plane

#### Rational Approximations

Cody and Hillstrom(1967) gives minimax rational approximations for ln Γ ( x ) for the ranges 0.5 ≤ x ≤ 1.5 , 1.5 ≤ x ≤ 4 , 4 ≤ x ≤ 12 ; precision is variable. Hart et.al.(1968) gives minimax polynomial and rational approximations to Γ ( x ) and ln Γ ( x ) in the intervals 0 ≤ x ≤ 1 , 8 ≤ x ≤ 1000 , 12 ≤ x ≤ 1000 ; precision is variable. Cody et.al.(1973) gives minimax rational approximations for ψ ( x ) for the ranges 0.5 ≤ x ≤ 3 and 3 ≤ x < ∞ ; precision is variable.

For additional approximations see Hart et.al.(1968) (Appendix B), Luke(1975) (pp. 22–23), and Weniger(2003) .

#### Expansions in Chebyshev Series

Luke(1969) gives the coefficients to 20D for the Chebyshev-series expansions of Γ ( 1 + x ) , 1 Γ ( 1 + x ) , Γ ( x + 3 ) , ln Γ ( x + 3 ) , ψ ( x + 3 ) , and the first six derivatives of ψ ( x + 3 ) for 0 ≤ x ≤ 1 . These coefficients are reproduced in Luke(1975) . Clenshaw(1962) also gives 20D Chebyshev-series coefficients for Γ ( 1 + x ) and its reciprocal for 0 ≤ x ≤ 1 . See Luke(1975) (pp. 22–23) for additional expansions.

#### Approximations in the Complex Plane

Rational approximations for Γ ( z + 1 ) A ( z ) , where A ( z ) = ( 2 π ) 1 2 ( z + c + 1 2 ) z + 1 exp ( - ( z + c + 1 2 ) ) , and approximations for Γ ( z + 1 ) based on the Padé approximants for two forms of the incomplete gamma function are in Luke(1969) . Luke(1975) (pp. 13–16) provides explicit rational approximations for ψ ( z ) + γ