All Questions
Tagged with complex-analysis continuity
387
questions
2
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0
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50
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Surface integral of a complex Log function
I am trying to calculate the surface integral of a complex Log function i.e.
$$ \int\int_{|z|<1}{Log(x+i y-(x_0+iy_0)))dxdy}$$
where $z=x+iy$ and $x_0,y_0 \in \mathbb{R}$ .
I know that for analytic ...
- 21
2
votes
1
answer
45
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Continuity of real part of a complex function
Consider a function $g(z)$ which is analytic on $\mathbb{C}_+$, and its range is contained in $\mathbb{C}_-$. Suppose that $g$ has a continuous extension to $\mathbb{C}_+ \cup \mathbb{R}$, denoted by $...
- 115
3
votes
1
answer
272
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Continuity of Hilbert transform
Suppose $f : \mathbb{R} \to \mathbb{R}$, be a non-negative, bounded and continuous function, and its support is a compact interval in $\mathbb{R}$. Moreover, we have that $\int f(x) \, dx =1$. The ...
- 115
0
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0
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89
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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
0
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86
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Continuity on boundary of convergence power series
I’m stuck trying to prove the following: Given $f(z) = \log(2+z^2)$, I consider its power series representation around $z=0$, which is $i(z) = \log(2)+\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n2^n}z^{2n}$...
- 1
5
votes
3
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124
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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
0
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0
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86
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Is a continuous bijection $f:\mathbb C \to \mathbb C$ a homeomorphism?
Let $X$ be an open subset of $\mathbb K$, $f: X \to \mathbb K$, and $Y:=f(X)$.
Theorem: If $\mathbb K = \mathbb R$, and $f$ is injective and continous, then $f^{-1}:Y \to X$ is continuous.
The proof ...
- 5,817
5
votes
1
answer
212
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How to show that any continuous map $f:\mathbb{C}^*\to \mathbb{C}^*$ is of the form $f(z)=z^me^{g(z)}?$
Given a nowhere vanishing continuous map $f:\mathbb{C}^*\to \mathbb{C}^*$, how do I show that there exist an integer $m\in \mathbb{Z}$ and a continuous map $g:\mathbb{C}^*\to \mathbb{C}$ such that $f$ ...
- 962
0
votes
1
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77
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Continuity and maxima of complex piecewise function
I need help showing the following:
Prove that the function
$$f:\mathbb{R}\to\mathbb{C},\quad f(t)=\begin{cases}e^{it},&t\geq0,\\1+it,&t<0,\end{cases}$$
is continuous everywhere.
I would ...
- 21
0
votes
1
answer
34
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Why $g$ is continuous on A?
I am trying to understand the proof of Schwarz Lemma below:
But I do not understand why $g$ is continuous on A? we do not know the formula for $f$ and so we do not know $f'(0),$ could someone explain ...
- 2,087
2
votes
1
answer
175
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Let $\gamma:[0,1] \to D = \mathbb C \setminus \{0 \}$ be a continuous closed curve. Show that $\gamma \approx \sigma$ in $D$ for..
Let $\gamma:[0,1] \to D = \mathbb C \setminus \{0 \}$ be a continuous closed curve. Show that $\gamma \approx \sigma$ in $D$ for some curve $\sigma$ whose trace is contained in $S^1$.
Hello, I ...
- 43
2
votes
1
answer
73
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Does the supremum norm $\|p\|_{A}$ depend continuously on subsets $A\subset\mathbb{C}$ with respect to the Hausdorff distance?
Consider the space $\mathcal{K}$ of all non-empty compact subsets of $\mathbb{C}$. One can show that the Hausdorff distance defined by
$$h(X,Y)=\max\bigg\{\sup_{x\in X}\inf_{y\in Y}|x-y|,\sup_{y\in Y}\...
- 3,376
0
votes
1
answer
70
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Verify that $u, \; v$ are continuous in a neighborhood of $z=0$ and satisfy the Cauchy-Riemann Eqns at $z=0$. Show that $f'(0)$ does not exist.
This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming qual. I'm really struggling to know where to go with it and any help would be ...
- 470
0
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1
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332
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definition of limits and continuity in complex analysis
Here is the definition my textbook gives:
Suppose a have a function $f$ with domain $\{z \in \mathbb{C} : |z| \le 1\}$. The point $i$ has $|i|=1$ and is in $f$'s domain. $f$ is not defined on any ...
- 77
1
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0
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156
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Can a probability generating function fail to be analytic on any neighborhood of 1?
Any probability generating function $G(z)$ is analytic on at least the open unit disk of the complex plane, because its Taylor series expansion about $z=0$ has a radius of convergence of at least 1. ...
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