Geodesic
Fast algorithms for elementary operations on complex power series
Diskretnaya Matematika, Tome 22 (2010) no. 1, pp. 17-49

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{DM_2010_22_1_a2,
     author = {I. S. Sergeev},
     title = {Fast algorithms for elementary operations on complex power series},
     journal = {Diskretnaya Matematika},
     pages = {17--49},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2010_22_1_a2/}
}
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I. S. Sergeev. Fast algorithms for elementary operations on complex power series. Diskretnaya Matematika, Tome 22 (2010) no. 1, pp. 17-49. http://geodesic.mathdoc.fr/item/DM_2010_22_1_a2/