Derivative of Uniform Limit of Analytic Functions

Theorem
Let $U$ be an open subset of $\C$.

Let $\left\{ {f_n}\right\}_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$.

Let $\left\{ {f_n}\right\}$ converge locally uniformly to $f$ on $U$.

Then the sequence $\left\{ {f'_n}\right\}_{n \mathop \in \N}$ converges locally uniformly to $f'$.

Proof
Note that by Uniform Limit of Analytic Functions is Analytic, $f$ is analytic.

Let $a\in U$.

Let $D$ be a disk of radius $r$ about $a$, contained in $U$.

We have Cauchy's Integral Formula for Derivatives


 * $\displaystyle f'_n \left({a}\right) = \frac 1 {2 \pi i} \int_{\partial D} \frac {f_n \left({z}\right)} {\left({z - a}\right)^2} \, \mathrm d z$

and:


 * $\displaystyle f'\left({a}\right) = \frac 1 {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)}{\left({z - a}\right)^2} \, \mathrm d z$

Therefore:

Now the $f_n$ tend uniformly to $f$, and we can bound $r$ away from zero.

It follows that $f_n' \to f'$ uniformly in each compact disk contained in $U$.

Also see

 * Uniform Limit of Analytic Functions is Analytic