All Questions
34
questions
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
$$
\begin{equation}
f(z)=
\begin{cases}
\...
- 11
0
votes
1
answer
83
views
Justification for approaching limit from any direction
I want to see a rigorous explanation why the following general fact:
$$ f \text{ continuous at z} \Longleftrightarrow \left( \forall (x_n)_{n \in \mathbb{N}} : \lim_{n \to \infty} x_n = z \implies \...
- 547
0
votes
1
answer
59
views
Continuity of a complex variable function.
Let $$ f(x)= \left\{ \begin{array}{lcc}
\frac{z^3-1}{z^2+z+1} & if & |z| \neq 1 \\
\\ \frac{-1+i\sqrt{3}}{2} & if & |z|=1 \\
\end{array}
\...
- 769
-1
votes
2
answers
114
views
Method for proving continuity for a complex function
We have that continuity for a complex function is defined as such: f is continuous at $z=z_0$ if it is defined in a neighborhood of $z_0$ and there exists a limit as:
\begin{equation}
\lim_{z\...
- 2,973
2
votes
0
answers
37
views
Showing Differentiability/Continuity at endpoints of closed interval?
I am given the function
$\gamma:[-1,\frac{\pi}{2}] \rightarrow \mathbb{C}$
$\gamma(t) =
\begin{cases}
t+1 & \text{for $-1 \leq t \leq0$} \\
e^{it} & \text{for $0 \leq t \leq\frac{\pi}{2}$} ...
- 296
0
votes
2
answers
1k
views
Prove that $f(z)$ is not continuous at $z=i$ where $f(z)=\frac{3z^4 -2z^3 +8z^2 -2z +5}{z-i}$
I tried to simplify it
$$f(z)=\frac{3z^4 -2z^3 +8z^2 -2z +5}{z-i}$$
$$f(z)=\frac{{3z^2 -2z +5}{z^2+1}}{z-i}$$
$$f(z)=\frac{(3z^2 -2z +5)(z+i)(z-i)}{z-i}$$
$$f(z)=(3z^2-2z+5)(z+i)$$
How to prove that ...
- 1
0
votes
1
answer
41
views
Finding the maximum of a complex valued function on a set
Let $R$ be the right half plane {$z ∈ \mathbb C : Re (z) > 0$}, and let $λ > 0$.
(a) Show that the function $e^{−λ/z}$ is holomorphic on $R$ and continuous on $\bar{R}$\ {$0$}.
(b) Does the ...
- 1
1
vote
1
answer
1k
views
Is $f(z)=z\sin(1/z)$ continuous for $z\rightarrow 0$?
I have $$f(z)=z*\sin(1/z)$$ for $z\neq 0$ and $$f(0)=0$$ to see whether the function is continuous or discontinuous at 0, $z\rightarrow 0$ where $z\in \mathbb{C}$. My first idea was to express the ...
- 1,073
1
vote
2
answers
248
views
Computing a limit of a very complicated complex exponential expression; Fourier Transform of $\tan(x)$
1. BACKGROUND
I stumbled upon this problem while doing something very unusual -- computing the Fourier Transform of the tangent function.
I got that:
$$\mathcal{F}\{\tan(x)\}=-i\cdot\Bigg(\int_{-\...
- 3,875
1
vote
2
answers
32
views
Limit in $ \mathbb{C}$. Can the function be defined in $ z_{0} = i +1$ so that it remains continuous?
Let $f \colon \mathbb{C} \backslash \left\{1+i\right\} \to \mathbb{C}$
$$ f(z) = \frac{z^2-2i}{z^2-2z+2}$$. Can the function be defined in $ z_{0} = i +1$ so that it remains continuous ?
I know I ...
- 1,632
1
vote
0
answers
38
views
Existence of the limit of a bounded analytic function
Let $X$ be a Banach space and let $U\subset\mathbb{C}
$ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\}
\rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{...
- 1,172
2
votes
1
answer
1k
views
Where is $\sqrt{z+1}$ analytic and continuous?
I am trying to determine where $$f(z)=\sqrt{z+1}$$ is analytic, where the square root is the principal branch.
I know that $\sqrt{w}$ is analytic for $\mathbb{C}\setminus(-\infty,0]$. So, I think $f(...
user557493
1
vote
0
answers
69
views
Show that $f : \mathbb R \to \mathbb C \in 2\pi\mathrm i \mathbb Z$ when $L : \mathbb C\setminus\{0\} \to \mathbb C$ is continuous
The Claim
Let $L : \mathbb C\setminus\{0\} \to \mathbb C$ be a continuous function and define $f : \mathbb R \to \mathbb C, t \mapsto L(\mathrm e ^{\mathrm i t}) - \mathrm i t$. Show that $f \in 2\...
- 701
0
votes
0
answers
28
views
$ g\in C^{2}(\Omega)$ are dense in $f\in C(\partial\Omega)$?
Let $\Omega$ a compact set in the complex plane.
How I can show that the space of $ g\in C^{2}(\Omega)$ with a compact support are dense in $f\in C(\partial\Omega)$?
- 1
0
votes
1
answer
42
views
Showing that a complex function is continuous
Find out, and give reason, whether $f(z)$ is continuous at $z = 0$ if $f(0) = 0$ and for $z\neq 0$ the function $f$ is equal to $(\operatorname{Re}z^2)/|z|$.
