Exercises
[1KV]Prerequisites:[1K9],[1DJ].Difficulty:*.
Consider the power series
\[ f(x)=β_{n=0}^β a_ n x^ n ~ ~ ,~ ~ g(x)=β_{m=0}^β b_ m x^ m~ , \]with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\). Suppose \(g(0)=0=b_ 0\). Let \(I_ f,I_ gβ β\) be disks centered in zero with radii less than \(r_ f\) and \(r_ g\), respectively: the previous series therefore define functions \(f:I_ fββ\) and \(g:I_ gββ\). Up to shrinking \(I_ g\), we assume that \(g(I_ g)β I_ f\).
Show that the composite function \(h=fβ¦ g:I_ gββ\) can be expressed as a power series \(h(x)=β_{k=0}^β c_ k x^ k \) (with radius of convergence at least \(r_ g\)). Show how coefficients \(c_ k\) can be computed from coefficients \(a_ k,b_ k\).
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