All Questions
12
questions
0
votes
1
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57
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In complex number system, sin z and cos z are unbounded and periodic. But they are continuous also. How can that be possible?
I know that a continuous periodic function must be bounded because if a function is continuous and periodic, its graph will have to turn at certain points to reattain the values and hence, it cannot ...
- 13
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
0
votes
1
answer
3k
views
If $f(z)$ is analytic at $z_0$ then it is continuous at $z_0$
My Proof: Since $f(z)$ is analytic at $z_0$ then it is differentiable at $z_0$
1) $f(z_0)$ exists because $f(z)$ is differentiable at $z_0$
2) $\lim_{z\rightarrow z_0}[f(z)-f(z_0)]=\lim_{z\...
- 375
1
vote
2
answers
157
views
Existence of continuous and analytic function
Let $A=\{z\in \Bbb C:|z|>1\}$; $B=\{z\in \Bbb C:z\neq 0\}$ Which of the following are true:
There is a continuous onto function $f:A\to B$
There is a continuous one-one function $f:B\to A$
There ...
- 31.2k
1
vote
1
answer
83
views
Is this a continuity / connectedness argument or is it an orientation-preservation argument?
Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle.
We know the fact that the upper half plane (UHP) ...
- 1
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
3
votes
1
answer
903
views
Analytic continuation of function continuous on boundary
Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|<1$. I wondered, can $f$ always be extended to a holomorphic function on some region ...
- 873
9
votes
7
answers
1k
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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
3
votes
1
answer
129
views
Relation between continuity of $f$ and analyticity of $f(z)^8$
If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
- 1,441
2
votes
3
answers
2k
views
let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant
$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant...
Then we are supposed to use this result to ...
- 95
3
votes
1
answer
807
views
To what extent is a function that is analytic on the unit disk determined by its boundary values?
Suppose we have a function that is analytic on the open unit disk.
Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate.
...
- 1,332
1
vote
2
answers
476
views
Is discontinuity along a line equivalent to branch cut?
Suppose I claim the analytic function $f(z)$ has a branch cut along the positive real line, how would one go on to prove this?
Is it sufficient to prove that $f(z)$ is discontinuous across this line?
...
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