All Questions
Tagged with complex-analysis continuity
387
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Prove that if $f(z)$ is continuous on closed region then it is bounded in that region
While reading text on complex analysis, I found a following question:
Question: Prove that if $f(z)$ is continuous on closed region then it is bounded in that region.
My attempt: Isn't the boundedness ...
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Complex analysis, Ian Stewart Exercise 4.7.5: Proving $\sqrt{z}$ is continuous on $\mathbb{C}\setminus\{x\leq0\}$
This is exercise 4.7.5 in Ian Stewart's "Complex Analysis
(The Hitch Hiker’s Guide to the Plane)":
Let $C_{\pi} =\{z\in\mathbb{C}:z\neq x\in\mathbb{R},x\leq0\}$ be the 'cut plane' with the ...
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Prove that $f(z)=\int_0 ^1 t^z dt$ is continuous
Let $$f(z)=\int_0 ^1 t^z dt.$$ Prove that $f$ is holomorphic on $\{\Re(z)>-1\}$.
My attempt: First notice that $$|t^z|=|e^{z\log(t)}|=e^{\Re(z\log(t))}=e^{\log(t)\Re(z)}=t^{\Re(z)},$$ and thus $$\...
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Isn't derivative of holomorphic function continuous?
On page 65 of Shakarchi's Complex analysis , problem 5 asks that if f is continuously complex differentiable on some set , under suitable conditions show that Goursat's theorem holds. He also advices ...
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Finding a region $G\subset\mathbb C$ such that $f,g$ defined on $G$ such that $f(z)^2= g(z)^2=1-z^2$ are continuous.
Find an open connected set $G\subseteq\mathbb{C}$ and two continuous functions $f,g$ defined on $G$ such that $f(z)^2=g(z)^2=1-z^2$. Can you make $G$ maximal? Are $f$ and $g$ analytic?
The following ...
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Continuity of maximum modulus function $M(r)=\max_{|z|=r}|f(z)|$
I am looking to prove that the maximum modulus function
$$M(r)=\max_{|z|=r}|f(z)|$$
is continuous on $[0, \infty)$ for $f$ an entire function.
My idea was to use the representation of $f$ as a power ...
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Why does this show Log can't be extended to whole $\mathbb{C}^*$
Why does the following show Log can't be extended to whole $\mathbb{C}^*$?
Here's another proof which I think I understand, though I'm not sure what's the connection between the two proofs:
I ...
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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 ...
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Weak hypothesis for Morera's theorem?
Morera's theorem states that : Let $f(z)$ is a continous in a domain $D$. If $\int_Cf(z)dz = 0$ for every simple closed contour lying in $D,$ then $f$ is analytic in $D$.
Does there exist a function $...
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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
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Proving that inverse of the unit circle parametrization is not continuous. [duplicate]
Statement:
Let us have a continuous and bijective unit-circle parametrization map:
$f: [0, 2\pi) \rightarrow S$
$\phi \mapsto cos(\phi) + i \cdot sin(\phi)$
We prove that $f^{-1}$ is not continuous.
...
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Justification for approaching limit from any direction
I want to see a rigorous explanation why the following general fact:
$$ f \text{ continuous at z} \Longleftrightarrow \left( \forall (x_n)_{n \in \mathbb{N}} : \lim_{n \to \infty} x_n = z \implies \...
- 547
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Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
- 1,861
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Identity theorem for (real) analytic functions on lower dimensional subsets
For simplicity, we will deal with $\mathbb{R}^2$. Let's assume we have an one-dimensional submanifold $M_1 \subset \mathbb{R}^2$ and two analytic function $F,G: M_1 \rightarrow \mathbb{R}$.
If I know $...
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Is there a simple way to interpolate smoothly between levels of a complex-valued continued fraction?
I have two complex numbers, $a = x_1 + y_1 i$ and $b = x_2 + y_2 i$. These serve as inputs to an infinite continued fraction of the form $f_n = a + \frac{b}{f_{n - 1}}$, with $f_1 = a$. Thus the first ...
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