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Exercises

  1. [1KV]Prerequisites:[1K9],[1DJ].Difficulty:*.

    Consider the power series

    \[ f(x)=βˆ‘_{n=0}^∞ a_ n x^ n ~ ~ ,~ ~ g(x)=βˆ‘_{m=0}^∞ b_ m x^ m~ , \]

    with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\). Suppose \(g(0)=0=b_ 0\). Let \(I_ f,I_ gβŠ‚ β„‚\) be disks centered in zero with radii less than \(r_ f\) and \(r_ g\), respectively: the previous series therefore define functions \(f:I_ fβ†’β„‚\) and \(g:I_ gβ†’β„‚\). Up to shrinking \(I_ g\), we assume that \(g(I_ g)βŠ‚ I_ f\).

    Show that the composite function \(h=fβ—¦ g:I_ gβ†’β„‚\) can be expressed as a power series \(h(x)=βˆ‘_{k=0}^∞ c_ k x^ k \) (with radius of convergence at least \(r_ g\)). Show how coefficients \(c_ k\) can be computed from coefficients \(a_ k,b_ k\).

    Solution 1

    [1KW]

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