All Questions
Tagged with complex-analysis continuity
388
questions
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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
$$
\begin{equation}
f(z)=
\begin{cases}
\...
- 11
1
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0
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13
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Continuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\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
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2
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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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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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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 $$(x,y)=\left(\cos(f_m (x,y)),\sin(f_m (x,y))\...
- 385
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0
answers
16
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Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints
It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
- 4,653
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44
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Understanding continuity in $\hat{\mathbb{C}}$
Let $\hat{\mathbb{C}}$ denote the Riemann sphere. Let $f:B_1(0) \to \hat{\mathbb{C}}$ be continuous. If $f$ is continuous at $z$ and non-zero, then $1/f(z)$ is continuous at $z$ as well. My question ...
- 730
2
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33
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Continuity of a function defined by an improper integral
Let $c > 0$ and let the function $f : (0, \infty) \to \mathbb{C}$ be defined as
$$
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
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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
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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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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
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1
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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
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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
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votes
2
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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
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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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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
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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
2
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1
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108
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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
0
votes
0
answers
36
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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 $...
- 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