All Questions
Tagged with complex-analysis continuity
389
questions
-3
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0
answers
22
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Continuity and Analyticity [closed]
Show that $f(z) = \sin x\cosh y+i\cos x\sinh y$ is continuous as well as analytic everywhere.
1
vote
0
answers
27
views
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
$$
\begin{equation}
f(z)=
\begin{cases}
\...
- 11
1
vote
0
answers
12
views
Continuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{...
- 3,263
0
votes
1
answer
55
views
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
61
views
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.
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 ...
- 2,522
1
vote
0
answers
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://...
- 996
0
votes
0
answers
43
views
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
0
votes
0
answers
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)...
- 996
0
votes
0
answers
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
5
votes
6
answers
628
views
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 \...
- 6,352
0
votes
1
answer
56
views
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 $$(x,y)=\left(\cos(f_m (x,y)),\sin(f_m (x,y))\...
- 385
0
votes
0
answers
16
views
Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints
It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
- 4,653
1
vote
0
answers
44
views
Understanding continuity in $\hat{\mathbb{C}}$
Let $\hat{\mathbb{C}}$ denote the Riemann sphere. Let $f:B_1(0) \to \hat{\mathbb{C}}$ be continuous. If $f$ is continuous at $z$ and non-zero, then $1/f(z)$ is continuous at $z$ as well. My question ...
- 730
2
votes
0
answers
33
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Continuity of a function defined by an improper integral
Let $c > 0$ and let the function $f : (0, \infty) \to \mathbb{C}$ be defined as
$$
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
0
votes
2
answers
128
views
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 ...
- 1,096