All Questions
16
questions
1
vote
0
answers
143
views
Continuity at the boundary of a convergent power series
Say $f(z)=\sum a_{n}z^{n}$ is a power series with convergence radius $0<R<\infty$. Suppose we know that the series convergence at $z_{0}$ where $z_{0}$ is a point at the boundary of the ...
- 1,501
0
votes
1
answer
112
views
Show that $\{b_n\}_{n \geq 1}$ is a Cauchy sequence.
Let $\Omega \subseteq \mathbb C$ be a region and $f : \Omega \longrightarrow \mathbb C$ be a continuous function. Let $\gamma : [0,1] \longrightarrow \Omega$ be a continuous path of bounded variation. ...
- 2,912
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
0
votes
2
answers
41
views
Why does this uniform continuity follow from continuity?
Let $K$ be a compact subset of $\mathbb{R}$. I'm reading a proof and it says the following:
Let $\epsilon > 0$. Since the mapping $\xi \mapsto e^{\imath \, \xi}$
is continuous, we can pick $\...
user745578
1
vote
0
answers
25
views
Prove that complex function is uniformly continous [duplicate]
$f: \mathbb{C} \rightarrow \mathbb{C}, \; z \mapsto f(z) := \frac{z^2}{1+|z|}$
How would I go about proving this is uniformly continuous? Currently I have only practically dealt with analyzing ...
- 33
1
vote
1
answer
91
views
Show the following for a uniformly continuous $f:\mathbb{C} \mapsto \mathbb{C}$
Show for a uniformly continuous $f:\mathbb{C} \mapsto \mathbb{C}$ that there exists $\alpha,\beta \in \mathbb{R}$ so that the following is true
$$\forall z\in\mathbb{C}: |f(z)| \le \alpha\cdot|z|+\...
- 129
1
vote
2
answers
458
views
Is the complex function $f(z)=\exp(-|z|)$ uniformly continuous on $\mathbb{C}$?
Is the complex function $f(z)=\exp(-|z|)$ uniformly continuous on $\mathbb{C}$?
The complex exponential function $\exp(z)$ is defined as
$$
\exp(z) := \lim_{n\to \infty} \Bigl(1+ \frac{z}{n}\Bigr)^...
- 129
2
votes
1
answer
100
views
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
I tried using the definition of the continuity to try and find a $\delta$ but I got stuck here
$$ |f(w) - f(z)| = |\frac{w^2 \cdot (1+|z|) - z^2 \cdot (1+|w|)}{(1+|w|)\cdot(1+|z|)}|$$
What can I do to ...
- 129
0
votes
0
answers
77
views
Confusion in the proof of Rouché's theorem (joint continuity part)
Let, $f,g$ be two holomorphic functions in a region $\Omega$. $C$ be a
circle in $\Omega$ containing interior such that $|f(z)|>|g(z)|\
\forall z\in C$. g vanishes nowhere on $C$. Then $f$ and $...
- 3,144
2
votes
3
answers
2k
views
How to prove that $\sqrt x$ is continuous in $[0,\infty)$?
I am trying to prove that
$\sqrt x$ is continuous in $[0,\infty)$.
I have started writing the following proof:
Given $x_0 \in [0,\infty)$ and $\epsilon > 0$. We have to show that there exists ...
- 1,053
4
votes
3
answers
2k
views
continuity of $e^{-1/z}$ in $\mathbb C$
Let f be the function defined by $f(z)=e^{-1/z}$ in $\mathbb C$, prove that $f$ is continuous in the set $0< \vert z \vert < 1$ and $\vert arg(z) \vert <\pi/2$ but it's not uniformly ...
- 659
0
votes
2
answers
2k
views
Uniform continuity of functions of a complex variable
Let $f:\color{blue}D→\mathbb{C}$ be a complex function with $\color{blue}D = \{z \in \mathbb{C} \:/\:\frac12\leq|z|\leq1 \}$
$$f(z) = \frac{1+z^2}{3+z^3}$$
1st I have to find all the discontinuities,...
- 2,425
1
vote
2
answers
2k
views
Is sum, product and composition of uniformly continuous functions also uniformly continuous?
I know that sum and composition are and product isn't but need prove of it or counter-example.
I have tried do it from definition but failed.
- 503
0
votes
1
answer
227
views
Limit at set boundary for uniform function
Let $D:=\{z\in \mathbb{C}: \lvert z \rvert < 1\}$ and $B := \{z\in \mathbb{C}: \lvert z \rvert = 1\}$. Let $f:D\to \mathbb{C}$ be a uniformly continuous function. Then $\lim\limits_{z\to z_0, z\in ...
- 9,708
1
vote
1
answer
1k
views
How to show that a complex-valued function is uniformly continuous?
should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function.
For example how can I proove that $f(x)=x^2,f:\mathbb{C}\longmapsto\...
