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We give new algorithms for the computation of square roots and reciprocals of power series in C x . If M (n) denotes the cost of multiplying polynomials of degree n, the square root to order n costs (1.333 . . .+o(1))M (n) and the reciprocal costs (1.444 . . .+ o(1))M (n). These improve on the previous best results, (1.8333 . . . + o(1))M (n) and (1.5 + o(1))M (n), respectively.doi:10.1090/s0025-5718-2010-02392-0 fatcat:umljxlbf2jc45honv7yesw6zei