EDB — 1KV

Exercises

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

   Consider the power series

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

   with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\). Suppose \(g(0)=0=b_ 0\). Let \(I_ f,I_ g⊂ ℂ\) be disks centered in zero with radii less than \(r_ f\) and \(r_ g\), respectively: the previous series therefore define functions \(f:I_ f→ℂ\) and \(g:I_ g→ℂ\). Up to shrinking \(I_ g\), we assume that \(g(I_ g)⊂ I_ f\).

   Show that the composite function \(h=f◦ g:I_ g→ℂ\) can be expressed as a power series \(h(x)=∑_{k=0}^∞ c_ k x^ k \) (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]↺↻