All Questions
Tagged with complex-analysis continuity
387
questions
-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
vote
0
answers
106
views
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
views
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
views
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:
$$\lim_{z\to0}\frac{...
- 4,653
0
votes
1
answer
57
views
How can this function be written without cases?
The function,
$$
f(x) = \begin{cases}
(e^x-1)/x & x\neq 0 \\
1 & x = 0
\end{cases}
$$
is continuous and differentiable at $x=0$. By composition, the $x\neq 0$ case is analytic everywhere ...
- 531
2
votes
0
answers
42
views
A function satisfying a condition is a polynomial of degree $\leq 1$
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function s.t $f(x)=\frac12(f(x+r)+f(x-r))$ for every $r>0, x\in\mathbb{R}$. Prove that $f$ is a polynomial of degree $\leq 1$.
This is a question ...
- 4,653
0
votes
1
answer
40
views
Extending $\frac{f(z)}{z}$ for holomorphic f to be continuous.
Define $f: D(0,1)-> C$ to be holomorphic such that$ f(0)=0$. I want to extend $\frac{f(z)}{z}$ to be continuous on $\overline D(0,r)$ for arbitrary $0<r<1$.
My initial guess was to define:
$...
- 74
0
votes
1
answer
59
views
Continuity of a complex variable function.
Let $$ 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
2
votes
1
answer
75
views
Checking my understanding of the identity theorem.
Suppose $f$ is an analytic function on a domain $\Omega$, $z_0\in\Omega$, $r>0$ and $D_r(z_0)=\{z\in\mathbb{C}| |z-z_0|<r\}\subset\Omega$.
Also, suppose $(f(z))^2=1$, $\forall z\in D_r(z_0)$.
...
- 65
4
votes
0
answers
106
views
Do both real and imaginary roots of a cubic equation need to continuous?
I have a cubic equation:
$X^3-UX^2-KX-L=0$ (1)
with $X=1-E+U$, $K=4(1-\gamma^2-\lambda^2)$, $L=4\gamma^2U$.
I solve Eq. (1) for the variable $E$ numerically for $U=2$ and different sets of parameter $\...
- 141
0
votes
2
answers
96
views
Complex Function Continuity
I am trying to get this through my head about continuity of complex functions.
Say you have $f(z) = \dfrac{z^2}{|z|}$, and I want to show that the function is continuous everywhere on $\mathbb{C}\...
- 73
1
vote
1
answer
125
views
Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$?
Is $\{ (x, y) \in \mathbb{C}^2 : x^2 + y^2 = 1\}$ connected in $\mathbb{C}^2$ ?
How do I approach this problem ?
I think that that the fact that continuous functions map connected sets to connected ...
1
vote
0
answers
143
views
Continuity at the boundary of a convergent power series
Say $f(z)=\sum a_{n}z^{n}$ is a power series with convergence radius $0<R<\infty$. Suppose we know that the series convergence at $z_{0}$ where $z_{0}$ is a point at the boundary of the ...
- 1,501
-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