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We have the following theorem for holomorphic functions.

Theorem (Weierstrass)

If $(f_n)$ is a sequence of holomorphic functions on an open set $U\subset\mathbb C$ such that $(f_n)$ tends uniformly to $f$ on every compact $K\Subset U$.

Then $f$ is also holomorphic and $({f_n}^{(k)})$ tends uniformly to $f^{(k)}$ for all $k$.

I would like to have a simple concrete application (without meromorphic functions, infinite products and elliptic functions) of this theorem if you know one. Thank you in advance.

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E. Joseph
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1 Answers1

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A corollary of Weierstraß' theorem is that for every normal family $\mathscr{F}\subset \mathscr{O}(U)$ (where normality is $\mathbb{C}$-normality, i.e. we don't allow sequences converging locally uniformly to $\infty$), the family

$$\mathscr{F}^{(k)} = \{ f^{(k)} : f \in \mathscr{F}\}$$

of $k^{\text{th}}$ derivatives is again normal.

This fact is occasionally useful (1, 2, 3).

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Daniel Fischer
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  • Thanks, though I'm looking for a more concrete example if possible. Your third link is very interesting nonetheless. – E. Joseph Oct 18 '16 at 21:34
  • We can use the theorem to prove that a power series is holomorphic on its disk of convergence. Though that can also be proven by calculations with power series and elementary estimates. It's more interesting if we take a locally uniformly convergent series of holomorphic functions, but the main application of that that I can think of now is the Mittag-Leffler theorem, and you didn't want meromorphic functions. – Daniel Fischer Oct 18 '16 at 21:38
  • @Yes indeed, because I'm reading a small course on the subject and meromorphic functions only come later. I'll keep looking for a concrete example, and if I find one I'll post it here. – E. Joseph Oct 18 '16 at 21:41
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    This one is similar to the third link, you may find it interesting too. I'll think about further (different) applications and add them to the answer if I find an interesting one. – Daniel Fischer Oct 18 '16 at 21:42