A note on the fast power series' exponential

Abstract

It is shown that the exponential of a complex power series up to order n can be implemented via (23/12+ o(1))M (n) binary arithmetic operations over C, where M (n) stands for the (smoothed) complexity of multiplication of polynomials of degree < n in FFT-model. Yet, it is shown how to raise a power series to a constant power with the complexity (27/8 + o(1))M (n).