All Questions
Tagged with complex-analysis continuity
388
questions
-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
3
votes
0
answers
116
views
Show continuity of Euler gamma function using DCT
The Euler Gamma function is defined by
$$\begin{align}
&\Gamma : \{z\in \mathbb{C} : \Re(z) > 0\} \rightarrow \mathbb{C} \\
&\Gamma(z) = \int_{0}^{\infty} x^{z-1}e^{-x}dx
\end{align}$$
My ...
- 31
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
0
answers
73
views
about the sufficient conditions for complex differentiability via Cauchy-Riemann:
I just noticed in the rule about the sufficient conditions for complex differentiability via Cauchy-Riemann:
When we're considering as to whether or not $g: \mathbb C \to \mathbb C$ is differentiable ...
- 13.7k
0
votes
1
answer
51
views
Determine derivative wherever the derivative exists of $-i(1-y^2)+(2x-y)(y)$
Is this correct? (Edit: I'm just going to outline the steps and post the rest as an answer.)
$g: \mathbb C \to \mathbb C, g(z) = -i(1-y^2)+(2x-y)(y)$
Step 1. $g$ is differentiable only on $\{y=x\}$.
...
- 13.7k
0
votes
0
answers
657
views
Cauchy-Riemann Equation satisfies at $z=0$
Let $f(z)=\begin{cases}
\frac{z^5}{\left | z \right |^4} & \text{ if } z\neq 0 \\ 0
& \text{ if } z=0
\end{cases} $
I could show this is continuous on $\mathbb{C}$.
And, I would like to show ...
- 1,288
1
vote
0
answers
51
views
Continuity of contour integral
Let $X$ be a Banach space, denote $S=\{z \in \mathbb{C}: 0 \leq \text{Re}(z) \leq 1\}$ and let $f:S \rightarrow X$ be a continuous function satisfying that $\|f\|_{\infty}= \sup_{z \in S}\|f(z)\|_X<...
- 411
3
votes
1
answer
536
views
continuous extension of holomorphic function up to the boundary
Denote $S=\{z \in \mathbb{C}|0 \leq \text{Re}(z) \leq 1\}$, let $X$ be a Banach space and let $f:S^\circ \rightarrow X$ be a holomorphic function. Under what assumptions does $f$ have a unique ...
- 411
0
votes
0
answers
79
views
Show that $|f(z)| \leq |z|$ for all $z ∈V$ for a holomorphic and continuous function
Let $V=D(0,1)$ is the unit disk. Suppose that $f ∈ C(\overline V) ∩ H(V)$ i.e. $f$ is continuous on $\overline V$ and $f\restriction_V$ is holomorphic on $V$, $f(0)=0$ and $|f(z)| \leq 1$ for all $z ...
- 39
0
votes
1
answer
113
views
Show that if $M ≥ 0$ and $|f(z)| ≤ M$ for all $z ∈ ∂V$ , then $|f(z)| ≤ M$ for all $z ∈ V $
Suppose that V is a bounded open subset of the plane and $f ∈ C(\overline V)
∩ H(V)$ i.e. $f$ is continuous on $\overline V$ and $f\restriction_V$ is holomorphic on $V.$
Show that if $M ≥ 0$ and $|f(z)...
- 61
2
votes
1
answer
218
views
$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.
Question: Suppose we have functions $f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.
My attempt: Consider function $h(z)=\frac{f(z)}{g(z)}$, ...
- 2,534
0
votes
1
answer
112
views
Show that $\{b_n\}_{n \geq 1}$ is a Cauchy sequence.
Let $\Omega \subseteq \mathbb C$ be a region and $f : \Omega \longrightarrow \mathbb C$ be a continuous function. Let $\gamma : [0,1] \longrightarrow \Omega$ be a continuous path of bounded variation. ...
- 2,912
3
votes
1
answer
117
views
Find $n$ so $f\left(z\right)=\begin{cases} \frac{\overline{z}^{n}}{z^{2}} & z\neq0\\ 0 & z=0 \end{cases}$ is continuous but not differentiable at $0$
I am given that:
$$f\left(z\right)=\begin{cases}
\frac{\overline{z}^{n}}{z^{2}} & z\neq0\\
0 & z=0
\end{cases}$$
is continuous at $0$, but not differentiable there and I need to find $n$. What ...
- 601
0
votes
2
answers
147
views
Could there be a complex function which is continuous everywhere and differentiable everywhere except at a single point?
If $U$ is an open set in $\mathbb{C}$ and $z_0\in U$, could there be a function $f:U\rightarrow \mathbb{C}$ which is continuous in $U$, differentiable in $U \setminus \{z_0\}$, but not differentiable ...
- 1,788
0
votes
1
answer
37
views
Example of a continuous function so that $v^2$ subharmonic and $v$ not subharmonic
I am trying to find an example of a continuous function $v:U(0,1)\rightarrow \mathbb{R}$ so that $v^2$ is subharmonic, but $v$ is not. I can't seem to find a match that satisfies both criteria,...
- 119