# §5.23 Approximations

## §5.23(i) Rational Approximations

Cody and Hillstrom (1967) gives minimax rational approximations for for the ranges , , ; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to and in the intervals , , ; precision is variable. Cody et al. (1973) gives minimax rational approximations for for the ranges and ; precision is variable.

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

## §5.23(ii) Expansions in Chebyshev Series

Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of , , , , , and the first six derivatives of for . These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for and its reciprocal for . See Luke (1975, pp. 22–23) for additional expansions.

## §5.23(iii) Approximations in the Complex Plane

See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of .

For rational approximations to see Luke (1975, pp. 13–16).