All Questions
16
questions
3
votes
1
answer
272
views
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
5
votes
3
answers
124
views
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
votes
1
answer
77
views
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
1
vote
0
answers
25
views
Prove that complex function is uniformly continous [duplicate]
$f: \mathbb{C} \rightarrow \mathbb{C}, \; z \mapsto f(z) := \frac{z^2}{1+|z|}$
How would I go about proving this is uniformly continuous? Currently I have only practically dealt with analyzing ...
- 33
0
votes
3
answers
310
views
Why Does a Function Extends Holomorphically when The Related Sum Converges?
I have seen some cases where to prove a function is holomorphic, it is proven that a sum derived from that function is convergent.
For example, in Newman's Short Proof of the Prime Number Theorem by ...
1
vote
0
answers
145
views
Step in the proof of the continuity of the Gauss Gamma function
Define the Gamma function as
\begin{align}
\Gamma(z):= \frac{1}{G(z)} \quad \forall \, z\in \mathbb{C}\setminus \{0, -1, -2, ... \}
\end{align}
where
\begin{align}
G: \mathbb{C} \rightarrow \mathbb{C}...
- 866
2
votes
3
answers
2k
views
How to prove that $\sqrt x$ is continuous in $[0,\infty)$?
I am trying to prove that
$\sqrt x$ is continuous in $[0,\infty)$.
I have started writing the following proof:
Given $x_0 \in [0,\infty)$ and $\epsilon > 0$. We have to show that there exists ...
- 1,053
1
vote
2
answers
59
views
Pointwise convergence but not uniformly convergence of $g_{n} \to 0$ when $g_{n}(x)=f\left(\frac{\sqrt[n]{x}}{1+ \sqrt[n]{x}}\right)$.
Let $f:[0,1] \to \mathbb{R}$ a continuous function such $f(0)=f(\frac{1}{2})=0$ and $(f[0, \frac{1}{2}]) \not \subset \lbrace 0 \rbrace$. For each $n \in \mathbb{N}$, let $g_{n}:[0, \infty ) \to \...
- 1,955
1
vote
0
answers
49
views
Let an analytic function $f$ have domain-set the punctured disk $\Delta^*(z_0,r)$. Assume the existence of a complex number $w_0$ and a sequence
Let an analytic function $f$ have domain-set the punctured disk $\Delta^*(z_0,r)$. Assume the existence of a complex number $w_0$ and a sequence $\{r_n\}$ of positive real numbers satisfying $r>r_1&...
- 485
0
votes
3
answers
405
views
Continuous function on the plane to real line.
I know a proof to show that the open disc is an open set in which I show the inclusion of a neighborhood into the disc. But a much shorter answer would be to find a continuous function from the plane ...
user264750
-2
votes
2
answers
206
views
Can $x^{p/q}$ be rigorously proven/disproven to be extended to a larger subset of reals that includes negative real numbers?
There have been numerous arguments that $x^{p/q}$, if $p/q$ is not an integer, should not be extended to a subset of reals that includes negative non-integers $x$-values. Many have concluded that this ...
- 1
2
votes
2
answers
114
views
Is $f(x)=\frac{|x|^2}{x}$ continuous?
$$f(x) =
\begin{cases}
\frac{|x|^2}{x}, & \text{if $x \neq 0$} \\
0, & \text{if $x=0$}
\end{cases}$$
Can someone please explain if f is continuous? Assume $x$ is a complex number
Hints ...
- 535
1
vote
2
answers
295
views
Prove the function to not be continuous at $z = 0$
$$f(3) = \begin{cases}
\dfrac{\mathrm{Re}(z)}{|z|} & \text{when $z \neq 0$} \\
0 & \text{when $z = 0$}
\end{cases}$$
Can someone please explain the concept behind solving such a problem? ...
- 113
2
votes
0
answers
193
views
Continuity in the complex plane
I was reading a book where it is claimed that a sufficient condition for
\begin{equation}
f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2
\end{equation}
to be continuous and is ...
- 477
0
votes
1
answer
73
views
Continuous complex function
I have this function
$$F(z)=\frac{1}{\alpha-i\sqrt{z}}$$ with $\alpha>0$ and the determination of the square root with $\Im z>0$. I have to study its continuity in the set
$$A=\lbrace z|a\leq\Re ...
- 1