grandes-ecoles 2022 Q16

grandes-ecoles · France · x-ens-maths-c__mp Not Maths Series convergence and power series analysis
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $g = f^\dagger$ the reciprocal series. Show that $\hat{g} \prec (1/\lambda)(I + \hat{F} \circ \hat{g})$, conclude using part C that $\rho(g) > 0$ if $\rho(f) > 0$.
We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $g = f^\dagger$ the reciprocal series. Show that $\hat{g} \prec (1/\lambda)(I + \hat{F} \circ \hat{g})$, conclude using part C that $\rho(g) > 0$ if $\rho(f) > 0$.