All Questions
Tagged with complex-analysis continuity
388
questions
10
votes
4
answers
2k
views
On continuity of roots of a polynomial depending on a real parameter
Problem
Suppose $f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]$ for all $t\in\mathbb R$, where $a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C$ are continuous on $t$...
- 9,803
9
votes
7
answers
1k
views
If $\,f^{7} $ is holomorphic, then $f$ is also holomorphic. [closed]
I need some help with this problem:
Let $ \Omega $ be a complex domain, i.e., a connected and open non-empty subset of $ \mathbb{C} $. If $ f: \Omega \to \mathbb{C} $ is a continuous function and $ ...
- 5,110
9
votes
3
answers
6k
views
Why is it clear from this formulation that f is continuous wherever it is holomorphic?
Hi I am new on here so not sure if this is right place to post but quick and presumably easy question:
So holomorphic at a point $z_0 \in \Omega$ is defined as the limit as $h\rightarrow 0$ of
$\frac{...
- 103
8
votes
4
answers
14k
views
Is $f(z)=\bar{z}$ continuous?
I have $z\in \mathbb{C}$, is $f(z)=\bar{z}$ continuous on the whole complex plane?
Note that $\bar{z}$ is the conjugate of $z\in \mathbb{C}$
I was thinking that if $z$ is on the real line, then $f(...
- 3,454
8
votes
1
answer
3k
views
Are bounded analytic functions on the unit disk continuous on the unit circle?
Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$.
Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e.
$$ \sup_{\varphi \in [0,2\pi]} |f(e^{i\...
- 990
8
votes
5
answers
3k
views
Show that $e^z$ is continuous on $\mathbb{C}$
I know that $e^z$ is continuous on $\mathbb{R}$, but how would I show this rigorously on $\mathbb{C}$ using the $\epsilon - \delta$ definition of continuity?
I know how to begin:
If $|z - z_0| < \...
- 1,092
8
votes
3
answers
1k
views
Continuous extension of analytic functions
Is it possible to prove the following statement or is there a counter-example:
Let $H=\{y>0\}$ be the upper half plane in the complex plane. If $f$ is an analytic function on $H$ and its real part ...
- 196
7
votes
1
answer
2k
views
Show that the map $\epsilon \to z_{\epsilon}$ is continuous
Suppose that $f$ and $g$ are holomorphic in a domain containing the unit disc $D=\{z| |z| \le 1 \}$. Suppose that $f$ has a simple zero at $z=0$ and vanishes nowhere in the unit disc.
Let $f_{\...
- 12.9k
6
votes
2
answers
2k
views
No continuous injective functions from $\mathbb{R}^2$ to $\mathbb{R}$
Which of the following statements is true?
$(a)$ There are at most countably many continuous maps from $\mathbb{R}^2$ to $\mathbb{R}$
$(b)$ There are at most finitely many continuous surjective maps ...
- 2,589
6
votes
1
answer
10k
views
Prove that $f(z)=z^2$ is continuous.
Prove that $f(z)=z^2$ is continuous for all complex and real values of $z$.
What I've got so far is:
Given $ \epsilon >0$ and $|z-z_0|<\delta$ after some calculations (which I've checked with ...
- 455
6
votes
1
answer
3k
views
If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $[-1,1]$ then is entire.
This is a problem from Complex Variable (Conway's book) 2nd ed.
(Section 4.4) 9. Show that if $f: \mathbb{C}\to\mathbb{C}$ is a continuous function such that $f$ is analytic off $[-1,1]$ then $f$ is ...
- 2,505
6
votes
2
answers
655
views
Strange definition of real differentiable function
I am currently refreshing my complex analysis knowledge (I had a course many years ago, but I've forgotten almost all of it). The German textbook I'm using, which is Funktionentheorie by Fischer & ...
- 27.9k
5
votes
6
answers
628
views
What is $\sqrt{-1}$? circular reasoning defining $i$.
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \...
- 6,620
5
votes
3
answers
124
views
Show that every complex number $c$ with $|c|\leq n$ can be written as $c=a_1+a_2+\cdots + a_n$ where $|a_j|=1$ for every $j$.
Let $n\ge 2$ be a positive integer. Show that every complex number $c$ with $|c|\leq n$ can be written as $c=a_1+a_2+\cdots + a_n$ where $|a_j|=1$ for every $j$.
I think one can come up with a ...
- 1,837
5
votes
2
answers
1k
views
Holomorphic function on unit disc has no continuous extension to boundary
The following is a qual study question:
Let $$f(z) = \sum_{n=1}^\infty \sqrt{n}z^n$$
Having proven that the radius of convergence is 1, I'm asked to show that this function cannot be extended to a ...
- 287