# §9.19 Approximations

## §9.19(i) Approximations in Terms of Elementary Functions

• Martín et al. (1992) provides two simple formulas for approximating to graphical accuracy, one for , the other for .

• Moshier (1989, §6.14) provides minimax rational approximations for calculating , , , . They are in terms of the variable , where when is positive, when is negative, and when . The approximations apply when , that is, when or . The precision in the coefficients is 21S.

## §9.19(ii) Expansions in Chebyshev Series

These expansions are for real arguments and are supplied in sets of four for each function, corresponding to intervals , , , . The constants and are chosen numerically, with a view to equalizing the effort required for summing the series.

• Prince (1975) covers , , , . The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).

• Németh (1992, Chapter 8) covers , , , , and integrals , , , (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of , , , , again to 15D.

• Razaz and Schonfelder (1980) covers , , , . The Chebyshev coefficients are given to 30D.

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

• Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of , stored at the nodes. and are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of , at the node. Similarly for , .

## §9.19(iv) Scorer Functions

• MacLeod (1994) supplies Chebyshev-series expansions to cover for and for . The Chebyshev coefficients are given to 20D.