I thought of this question:

$f, g$ are functions that could be written as a power series in a disc with radius $R>0$, i.e. $$\forall |x|<r, f(x)=\sum a_n x^n, g(x)=\sum b_n x^n$$. If $b_0 = 0$, do we always have that $f(g(x))$ could be written as a power series in a disc containing 0?

I can't prove or find a counterexample, could anyone give me a hand?

Thanks a lot~

  • $\begingroup$ Is some complex analysis acceptable, or do you want a real-methods-only proof? $\endgroup$ – Daniel Fischer May 27 '17 at 13:52
  • $\begingroup$ @DanielFischer I'd like to be in a complex plane :) $\endgroup$ – pqros May 27 '17 at 14:00
  • $\begingroup$ Then: the composition of holomorphic functions is holomorphic; and a function is holomorphic if and only if it is (complex) analytic. $\endgroup$ – Daniel Fischer May 27 '17 at 14:02
  • $\begingroup$ We can express the composition of $f (g (x)) $ as a *formal power series* when $b_0=0$, but this does not tell us the radius of convergence for the result. $\endgroup$ – hardmath May 27 '17 at 14:06
  • $\begingroup$ @DanielFischer Oh I think I should reread my complex analysis, thanks~ $\endgroup$ – pqros May 27 '17 at 14:09

$f (x) $ exists if $|x|<R $

$f (g (x)) $ exists if $|g (x)|<R $.

Since $g (0)=0$, there exists $\eta>0$ such that

$g ((-\eta,\eta))\subset (-R,R) $.

thus $f (g (x)) $ exists for $|x|<\eta. $


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