in complex analysis need some examples about uniform convergence and analyticity

Question

I am new in solving anything in complex and I am stuck on two examples :

1) I read that : The limit of the sequence of non-analytic functions converging uniformly inside a simple closed curve can be analytic !! but i can't find an example.

and

2)a sequence (or series) of functions that converges uniformly on all compact subsets of a domain can be not-uniform throughout the domain!!

Answer 1

1. Let $$f_n(z) := \frac{1}{n} \cdot \bar{z}$$ for $z \in U := \{w \in \mathbb{C}; |w|<1\}$. Then $f_n$ is non-analytic (since $z \mapsto \bar{z}$ is non-holomorphic, hence non-analytic) and $f_n \to 0$ uniformly on $U$. And obviously $z \mapsto 0$ is analytic.

   In fact one can generalize this: Let $f$ a arbritary non-analytic bounded function on some domain $U$ and define $f_n$ on $U$ by $$f_n := \frac{1}{n} \cdot f$$ Then still $f_n$ is non-analytic and $f_n \to 0$ uniformly on $U$.

2. Let $$f_n(z) := \sum_{k=0}^n \frac{z^k}{k!}$$ for $z \in U:=\mathbb{C}$. Then $f_n \to \exp$ uniformly on compact subsets, but $f_n$ does not converge uniformly to $\exp$ on $\mathbb{C}$. In this example the main point is the unboundedness of the domain $U$.

   There are also examples where $U$ is bounded and still the sequence does not converge uniformly, for example $$f_n(z) := z^n$$ for $z \in U:=\{w \in \mathbb{C}; |w| <1\}$. Then $f_n \to 0$ on compact subsets of $U$, but not uniformly on $U$. (Note that this is example is similar to one of the standard examples in real analysis showing that pointwise convergence does not imply uniform convergence.)