We shall use the r-method of Lanczos to obtain rational approximations to the logarithm which converge in the entire complex plane slit along the nonpositive real axis, uniformly in certain closed regions in the slit plane. We also provide error estimates valid in these regions.1. Suppose z is a complex number, and s(l, z) denotes the closed line segment joining 1 and z. Let y be a positive real number, y ¿¿ 1. As a polynomial approximation, of degree n, to log x on s(l, y) we propose p* G Pn, Pn being the set of polynomials with real coefficients of degree at most n, which minimizes Thus, minimizing \\xp' -1|| subject to p(l) = 0 is equivalent to minimizing ||«-|| among all i£P" which satisfy -ir(0) = 1. This, in turn, is equivalent to maximizing |7r(0) I among all v G Pn which satisfy ||ir|| = 1, in the sense that if vo maximizes |7r(0)[ subject to ||x|| = 1, then** = -iro/-7ro(0) minimizes ||x|| subject to -ir(0) = 1. For, -v*(0) = 1 and since ||7r0|| = 1, ||ir*|| = l/|ir0(0)|. Now suppose v G Pn satisfies -ir(0) = 1, then vi = 7r/||ir|| has norm 1 and so |iri(0)| = |ir(0)|/||7r|| = l/lk|| = |tto(0)| = l/||.r*||, from which we conclude that ¡|tt*|| = ||t||.Since 0 is exterior to s(l, y), it is known (cf. Rivlin and Shapiro [3, p. 684]) that the Chebyshev polynomial for s(l, y), Tn.y(x) = Tn((2x -(1 + V))/(l -V)) , (where T"(x) = cos nd, x = cos 9, 0 = 8 _ v) is the polynomial satisfying || r",v|| = 1 and |*(0)| < |.Tn,.,(0)| for all v G Pn, \\v\\ = 1, v ^ ± Tn,y. Thus v" is Tn,y and ||xp' -1H is minimized, subject to p(l) = 0 for p = p* when
