All Questions
Tagged with complex-analysis continuity
387
questions
1
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27
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Determine where piecewise function is analytic and differentiable
The following is Problem 6.1 from a book I'm self-studying, the "Mathematics of Classical and Quantum Physics", by Byron and Fuller, 1e. Given
$$
\begin{equation}
f(z)=
\begin{cases}
\...
- 11
1
vote
0
answers
13
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Continuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{...
- 3,263
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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
2
votes
2
answers
66
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Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property. [duplicate]
Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
- 2,612
1
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0
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52
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Proof that a homeomorphism map boundaries to boundaries
I want to prove that if I have two topological spaces $X$ and $Y$, with $A \subset X$, and a homeomorphism $f : X \to Y$, then $f(\partial A) = \partial \big(f(A)\big)$.
I saw a proof here: https://...
- 1,021
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0
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43
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What can we say about $f$ and $g$?
Suppose $f$ and $g$ are holomorphic on a bounded domain $D$ and continuous on $\bar D$. Suppose also $|f(z)|=|g(z)|\neq0$ on $\partial D$ and $\frac{|g(z)|}{3}\leq|f(z)|\leq 3|g(z)|$ for all $z\in D$. ...
- 191
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39
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Question about proof of Lindelöf Theorem
Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)...
- 1,021
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62
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What is Hurwitz Theorem and how is it applied?
Dobner in his paper defines (https://arxiv.org/abs/2005.05142) some complicated function $\Phi_F$ (Eq. 8) and then a page after defines $H_t(z) = \int_{-\infty}^{\infty} e^{tu^2} \Phi_F(u) e^{izu} du$...
- 281
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6
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628
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What is $\sqrt{-1}$? circular reasoning defining $i$.
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \...
- 6,620
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1
answer
57
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Continuity Of Argument Function.
Fix $m\in \mathbb R$.
Define $f_m :\mathbb R^2 \setminus\{(0,0)\}\rightarrow(m,m+2\pi]$
$~~$as $(x,y) \mapsto$ argument of $(x,y)$ in $(m,m+2\pi]$.
i.e $$(x,y)=\left(\cos(f_m (x,y)),\sin(f_m (x,y))\...
- 385
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0
answers
16
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Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints
It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
- 4,653
1
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0
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44
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Understanding continuity in $\hat{\mathbb{C}}$
Let $\hat{\mathbb{C}}$ denote the Riemann sphere. Let $f:B_1(0) \to \hat{\mathbb{C}}$ be continuous. If $f$ is continuous at $z$ and non-zero, then $1/f(z)$ is continuous at $z$ as well. My question ...
- 730
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33
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Continuity of a function defined by an improper integral
Let $c > 0$ and let the function $f : (0, \infty) \to \mathbb{C}$ be defined as
$$
f(y) = \int_{c - i\infty}^{c+i\infty} \frac{y^s}{s(s+1)} \, ds.
$$
I want to show that $f$ is continuous.
My ...
- 163
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2
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128
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As a matter of fact, it is impossible to find a continuous $f$ such that $(f(z))^2=z$ for all $z$. ("Calculus Fourth Edition" by Michael Spivak.)
As a matter of fact, it is impossible to find a continuous $f$ such
that $(f(z))^2=z$ for all $z$. In fact, it is even impossible for
$f(z)$ to be defined for all $z$ with $|z|=1$.
To prove this by ...
- 1,138
0
votes
1
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62
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To justify a complex-valued function is continuous
A complex-valued function is defined on the unit disk as $f(z) = \int_{0}^{1} \frac{1}{1-tz} dt$. How can we show that the function is continuous ?
My Approach: As the integrand is analytic in $z$, it ...
- 379
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56
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Prove that if $f(z)$ is continuous on closed region then it is bounded in that region
While reading text on complex analysis, I found a following question:
Question: Prove that if $f(z)$ is continuous on closed region then it is bounded in that region.
My attempt: Isn't the boundedness ...
3
votes
1
answer
70
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Complex analysis, Ian Stewart Exercise 4.7.5: Proving $\sqrt{z}$ is continuous on $\mathbb{C}\setminus\{x\leq0\}$
This is exercise 4.7.5 in Ian Stewart's "Complex Analysis
(The Hitch Hiker’s Guide to the Plane)":
Let $C_{\pi} =\{z\in\mathbb{C}:z\neq x\in\mathbb{R},x\leq0\}$ be the 'cut plane' with the ...
- 451
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1
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66
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Prove that $f(z)=\int_0 ^1 t^z dt$ is continuous
Let $$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)},$$ and thus $$\...
- 146
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0
answers
48
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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 ...
- 477
2
votes
1
answer
192
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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 ...
- 11.5k
1
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1
answer
256
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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 ...
- 5,611
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votes
0
answers
78
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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 ...
- 451
3
votes
2
answers
85
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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 ...
- 3,376
2
votes
1
answer
84
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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 $...
- 751
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1
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132
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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
1
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2
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94
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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.
...
- 481
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1
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83
views
Justification for approaching limit from any direction
I want to see a rigorous explanation why the following general fact:
$$ f \text{ continuous at z} \Longleftrightarrow \left( \forall (x_n)_{n \in \mathbb{N}} : \lim_{n \to \infty} x_n = z \implies \...
- 547
2
votes
1
answer
108
views
Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
- 1,861
0
votes
0
answers
36
views
Identity theorem for (real) analytic functions on lower dimensional subsets
For simplicity, we will deal with $\mathbb{R}^2$. Let's assume we have an one-dimensional submanifold $M_1 \subset \mathbb{R}^2$ and two analytic function $F,G: M_1 \rightarrow \mathbb{R}$.
If I know $...
- 661
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1
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103
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Is there a simple way to interpolate smoothly between levels of a complex-valued continued fraction?
I have two complex numbers, $a = x_1 + y_1 i$ and $b = x_2 + y_2 i$. These serve as inputs to an infinite continued fraction of the form $f_n = a + \frac{b}{f_{n - 1}}$, with $f_1 = a$. Thus the first ...
- 1,861
2
votes
0
answers
50
views
Surface integral of a complex Log function
I am trying to calculate the surface integral of a complex Log function i.e.
$$ \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
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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
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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
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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
views
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
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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}$$
is continuous everywhere.
I would ...
- 21
0
votes
1
answer
34
views
Why $g$ is continuous on A?
I am trying to understand the proof of Schwarz Lemma below:
But I do not understand why $g$ is continuous on A? we do not know the formula for $f$ and so we do not know $f'(0),$ could someone explain ...
- 2,087
2
votes
1
answer
175
views
Let $\gamma:[0,1] \to D = \mathbb C \setminus \{0 \}$ be a continuous closed curve. Show that $\gamma \approx \sigma$ in $D$ for..
Let $\gamma:[0,1] \to D = \mathbb C \setminus \{0 \}$ be a continuous closed curve. Show that $\gamma \approx \sigma$ in $D$ for some curve $\sigma$ whose trace is contained in $S^1$.
Hello, I ...
- 43
2
votes
1
answer
73
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Does the supremum norm $\|p\|_{A}$ depend continuously on subsets $A\subset\mathbb{C}$ with respect to the Hausdorff distance?
Consider the space $\mathcal{K}$ of all non-empty compact subsets of $\mathbb{C}$. One can show that the Hausdorff distance defined by
$$h(X,Y)=\max\bigg\{\sup_{x\in X}\inf_{y\in Y}|x-y|,\sup_{y\in Y}\...
- 3,376
0
votes
1
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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
votes
1
answer
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
vote
0
answers
156
views
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. ...
- 6,280
-1
votes
2
answers
101
views
Proving the principal argument not continuous using standard metrics [closed]
Let $\operatorname{Arg}: \Bbb{C} \setminus \{0\} \to\Bbb{R}$ be the principal value of the argument, taking values in $(−\pi, \pi]$. Using the standard metrics on $\Bbb{C} \setminus \{0\}$ and $\Bbb{R}...
- 19
1
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0
answers
106
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Determining the Image of a Conformal Mapping
I'm having some trouble rigorously determining the images of conformal mappings in practice. As Marsden and Hoffman explain in their book Basic Complex Analysis, it suffices to analyze boundaries, as ...
- 2,221
2
votes
1
answer
218
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Continuity of $\text{Im}\frac{z}{z-1},\frac{\text{Re }z}{z},\text{Re }z^2,\frac{z\text{Re }z}{\left|z\right|}$
Find the set of points for which the given functions are continuous on that points.
$\text{Im}\frac{z}{z-1}$
$\frac{\text{Re }z}{z}$
$\text{Re }z^2$
$\frac{z\text{Re }z}{\left|z\right|}$
My ...
- 69
3
votes
1
answer
114
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Existence of Continuous Function on a Complex Region
I am working on the following problem:
Let $\Omega = \mathbb C\backslash [-1,1]$, i.e. deleting ``the line'' only, is there a function $f:\Omega\to \mathbb C$ such that $f$ satisfies $f(z)^2 = 1-z^2$ ...
- 1,042
1
vote
1
answer
48
views
Proving a function is holomorphic in $\mathbb{C}\setminus[0,1]$
Let $h(t):[0,1]\to\mathbb C$ be a continuous function. Prove that $f:\mathbb C\setminus[0,1]\to\mathbb C $ defined by $f(z)=\int_0^1\frac{h(t)}{z-t}$ is holomorphic.
My attempt:
$$\lim_{z\to0}\frac{...
- 4,653