All Questions
25
questions
0
votes
1
answer
70
views
Verify that $u, \; v$ are continuous in a neighborhood of $z=0$ and satisfy the Cauchy-Riemann Eqns at $z=0$. Show that $f'(0)$ does not exist.
This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming qual. I'm really struggling to know where to go with it and any help would be ...
- 470
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
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
1
answer
115
views
Prove that $f(z)=\begin{cases} \frac{ \overline{z}^3}{z^2} & z \neq 0\\0 & z=0\end{cases}$ is continuous at $0$ but $f'(0)$ doesn't exist
$$f(z)=\begin{cases} \frac{ \overline{z}^3}{z^2} & z \neq 0\\0 & z=0\end{cases}$$
prove that $f(z) $ is continuous at $z=0$ but $f'(0)$ doesn't exist.
I think maybe using Cauchy-Riemann ...
0
votes
0
answers
32
views
A question regarding differentiability and the boundary of analytic functions
Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
- 852
0
votes
1
answer
59
views
Given $\gamma(t)$ a $\mathcal{C}^1$ path on $\mathbb{C}^{\times}$, why is it true that $|\gamma(t)|$ is also $\mathcal{C}^1$?
Suppose $\gamma:[0,1]\to\mathbb{C}^{\times}$ is continuously differentiable, there is a proof I saw involved using the fact that $s(t)=|\gamma(t)|$ must also continuously differentiable but I cannot ...
- 3,886
2
votes
0
answers
441
views
Example of Looman-Menchoff theorem
I would like to know of one example of a function that strictly satisfies the Loomann-Menchoff theorem.
Namely, I want a complex function $f=u+iv$, defined on some $A \subseteq \mathbb C$, such that
...
- 2,588
0
votes
1
answer
44
views
Prove that a function is continuous at a point
Prove that the function f(z) defined by $$f(z)=\frac{x^3(1+i)-y^3(1-i)}{x^2 + y^2} ; z\neq{0};f(0)=0$$
is continuous and the Cauchy Riemann equations are satisfied at the origin $(0,0)$, yet $f'(0)$ ...
2
votes
1
answer
475
views
For complex function : $f'(z)$ exists $\implies$ $f$ continous in $z$?
Consider a complex function $f(z): A\subset\mathbb C \to\mathbb C$.
If the derivative of $f$ exists then $f$ must necessarily be a continuous function?
Is the following true? $f'(z)$ exists $\...
- 205
1
vote
1
answer
59
views
On uniformly continuous, bounded, hermitian functions whose product is continuously differentiable
Let $f,g: \mathbb R \to \mathbb C$ be uniformly continuous, such that $f(0)=g(0)=1$, $|f(x)|\le 1, |g(x)|\le 1,\forall x\in \mathbb R$, $f(-t)=\overline {f(t)}, g(-t)=\overline {g(t)}, \forall t \in \...
- 3,705
3
votes
1
answer
118
views
Where is $k(z)=PV(z-1)^{\frac{1}{2}}PV(z+1)^{\frac{1}{2}}$ Continuous and Differentiable
Consider the function ($PV$ denotes the principal value) $$PV(z-1)^{\frac{1}{2}}PV(z+1)^{\frac{1}{2}}, \ \ \forall z\in\mathbb{C}.$$
Find where $k$ is continuous and differentiable, giving reasons.
...
- 617
0
votes
1
answer
93
views
What does a function look like if its derivative is not continuous? [closed]
In a previous question about the Cauchy-Riemann condition in complex analysis, I learned that a function can have derivatives in a region, but its derivatives might not be continuous.
My question is:
...
- 1,201
3
votes
1
answer
1k
views
Existence of partial derivatives & Cauchy-Riemann does not imply differentiability example
I learned about the Cauchy-Riemann equations today, and my instructor used the following example to show that differentiability is not guaranteed if the partial derivatives are not continuous.
Let
$$
...
- 33