All Questions
6
questions
2
votes
2
answers
66
views
Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property. [duplicate]
Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
- 2,612
3
votes
2
answers
85
views
Is a holomorphic $f\colon U\to\mathbb{C}$ with continuous extension to $\overline{U}$ Lipschitz continuous on $\partial U$?
Let $U\subset\mathbb{C}$ be a bounded connected open subset with smooth boundary $\partial U$. Suppose that we have a holomorphic function $f\colon U\to\mathbb{C}$ that can be continuously extended to ...
- 3,376
3
votes
1
answer
132
views
Can this given $f: S^1\to \mathbb C$ be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C, F$ is holomorphic on $\mathbb D$?
Suppose that $f: \mathbb S^1\to \mathbb C$ is continuous such that $f(z)=f(\bar z)$ for all $z\in \mathbb S^1$. Can it be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C$ such that $ F$ ...
- 11.5k
1
vote
1
answer
128
views
Doubt about integral Cauchy theorem proof
I'm having trouble with the proof of the homological integral Cauchy theorem. I'm studying on Serge Lang, complex analysis, chapter $4$, page $148$, theorem $2.5$.
$f\colon A \subseteq \mathbb{C} \to \...
- 711
1
vote
0
answers
52
views
Prove that $F_{m}$ is continuous.
I'm trying to verify that $F_{m}$ it is more continuous, but I'm not getting it. I've tried using Newton's Binomial, but it did not..
Let $\lambda$ be a rectifiable curve and suppose $\phi$ is a ...
- 806
2
votes
1
answer
69
views
Proof $\lim_{r\to 0}\int_{C_r} f(z)dz=iA\alpha$
I have to prove the following assertion:
If $f(z)$ is continuous in the sector $0<\vert z-a\vert \le r_0, 0\le arg(z-a)\le \alpha ,(0<\alpha \le 2\pi)$ and the limit $\lim_{z\to a}[(z-a)f(z)]=A$...
- 1,658