Suppose we have a real entire function $f(x)$, so that real power series $$f(x) = \sum_{k=0}^\infty a_k x^k$$ converges for all $x\in\mathbb{R}$, and it can be analytically extended to an entire function $f:\mathbb{C}\to\mathbb{C}$.

If a sequence $(f_k)$ of real entire functions converges to $f$ uniformly on compact subsets of $\mathbb{R}$, is it true that $$f_k^{(n)}(0)\to f^{(n)}(0)$$ for any positive integer $n$?

  • $\begingroup$ Apply the Cauchy integral formula for the derivative. $\endgroup$ – LutzL Nov 12 '17 at 9:52
  • $\begingroup$ @LutzL That would be great if we were given uniform convergence on compact subsets of $\mathbb C$. But we're only given uniform convergence on compact subsets of $\mathbb R$. $\endgroup$ – David C. Ullrich Nov 12 '17 at 14:40

No. Let $f_k(x) = \sin(k^2x)/k$. Then $f_k\to0$ uniformly on $\mathbb R$, but $f_k'(0)=k$.


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