Rational Chebyshev approximations to the psi (digamma) function are presented for .5 á x S 3.0, and 3.0 ¿ x. Maximum relative errors range down to the order of 10-20. 1. Introduction. The principal mathematical properties of the psi (digamma) function (1) *(z) = dlln Tiz)]/dz = T'iz)/Tiz) are summarized by Davis [2] and Luke [3]. For real arguments, the function is traditionally computed using either the classical power series expansion CO (2) HI + z) = -7 + Z i-min)zn-\ \z\ < 1, or the

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... ic expansion (3) ¿(z) ~ ln(z) -f -£ -% , 2z ~l 2nz with the recurrence relation (4) iKz + 1) = iKz) + 1/z. The reflection formula (5) iKl -z) = xyiz) + x cotOrz) allows computation for negative arguments. (For complex arguments, see Luke [4].) Recently, Luke [3] presented an expansion of \pix + 3), 0 i£ x ^ 1, in Chebyshev polynomials, 17 coefficients being required to compute the function with an absolute error on the order of 10"20. For computations outside of the primary range, it is still necessary to use one or more of the relations (3), ( 4 ) and ( 5 ) in addition to Luke's expansion. In this note, we present rational Chebyshev approximations which allow direct computation of \pix) for any x ^ .5 with various choices of maximum relative error, including some of the order of 10~20. For x < .5, either (4) or ( 5 ) is still required in conjunction with our approximations.