All Questions
15
questions
0
votes
0
answers
16
views
Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints
It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
- 4,653
2
votes
1
answer
192
views
Finding a region $G\subset\mathbb C$ such that $f,g$ defined on $G$ such that $f(z)^2= g(z)^2=1-z^2$ are continuous.
Find an open connected set $G\subseteq\mathbb{C}$ and two continuous functions $f,g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$. Can you make $G$ maximal? Are $f$ and $g$ analytic?
The following ...
- 11.5k
0
votes
0
answers
89
views
Inverse of a analytic function
Let $f$ be a map on the closed unit disc $\bar{\mathbb{D}}$ in $\mathbb{C}$ such that $f$ is analytic on $\mathbb{D}$ and continuous on $ \bar{\mathbb{D}}$. Can you tell under what condition will $f$ ...
- 386
1
vote
3
answers
139
views
How to show that a continuous function analytic in a deleted neighbourhood is analytic in the entire neighbourhood?
Suppose
$$
F(z)=
\begin{cases}
\dfrac{f(z)-f(a)}{z-a}, & z\ne a\\[2ex]
\quad f'(a), & z=a
\end{cases}
$$
Here $f$ is analytic on a simply connected domain $D$ containing $a$. Clearly, $F$ is ...
- 594
0
votes
0
answers
115
views
Reason(s) why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$
I am studying complex analysis and I am confused about why $f(z)=1/(z-z_0)$ is not analytic at $z=z_0$. For a function to be analytic, it must be differentiable and single-valued. Obviously, $f'(z)=-1/...
- 39
0
votes
0
answers
32
views
A question regarding differentiability and the boundary of analytic functions
Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
- 852
1
vote
2
answers
118
views
If $f$ is a mapping from $\overline{B_1(0)}$ to itself that is continuous and analytic, prove that $f$ has a fixed point.
I am new to this but I kept on looking answer about this problem
Suppose $\displaystyle{f: \overline{B_1(0)} \rightarrow \overline{B_1(0)}}$ is continuous and $f$ is analytic in $\displaystyle{B_1(0)}$...
1
vote
1
answer
376
views
Complex function with values on the unit circle copied everywhere
If $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ is a function such that $f(z)=f(\frac z{|z|})$ and its restriction to unit circle is continous,then
$(1)\lim _{z\to 0} f(z)$ exist.
$(2)f$ is analytic ...
- 2,589
0
votes
1
answer
244
views
Holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$
Does there exist a holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$ where $ D= \{ z \in \mathbb{C} : |z|<1\}$.
I cannot use any of Schwarz lemma or ...
- 1,132
1
vote
2
answers
68
views
Analyticity and continuity in $\mathbb C$ of $f(z)=\frac{ \bar{z}^2}{z} $ if $z \ne0$,$0$ if $z=0$
For $z \in \mathbb C$
$f(z)=\frac{ \bar{z}^2}{z} $ if $z \ne0$,$0$ if $z=0$
I am investigating its analyticity and continuity in $\mathbb C$ or in any open neighbourhood of zero.
I know a complex ...
- 39
1
vote
0
answers
38
views
Existence of the limit of a bounded analytic function
Let $X$ be a Banach space and let $U\subset\mathbb{C}
$ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\}
\rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{...
- 1,172
0
votes
1
answer
83
views
Power series of analytic function
Suppose $f(z)=\sum\limits_{n=0}^{\infty}a_n z^n$ for $|z|<r$, where $r>0$.
Also $f(z)$ is continuous on $|z|\leq r$.
My question is whether the function $f(z)$ is analytic in some "big" disk $|z|...
- 8,455
1
vote
2
answers
3k
views
Necessary and Sufficient Condition for Analyticity of a Polynomial
Definition: A polynomial $P(x,y)$ will be called an analytic polynomial if there exist (complex) constants $a_k$ such that $$P(x,y) = a_0 + a_1 (x+iy) + a_2 (x+iy)^2 + ... + a_n(x+iy)^n$$
Definition: ...
- 8,040
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
2
votes
2
answers
114
views
A question regarding complex valued function
Let $A=\{z\in \mathbb C:|z|>1\}$ and $B=\{z\in \mathbb C:z\neq 0\}$. Then which of the following is/are true?
There exists a continuous onto function $f:A\to B$.
There exists a continuous one-one ...
- 4,928