Fast algorithms for elementary operations on complex power series

It is shown that the inversion of a complex-valued power series can be realised asymptotically with complexity of 5/4 multiplications (if we compare the upper bounds). It is shown that the calculation of the square root requires asymptotically also no more than 5/4 multiplications, the computation of an exponential has the complexity equal to 13/6 multiplications, and raising to an arbitrary power requires 41/12 multiplications.