All Questions
8
questions
8
votes
5
answers
3k
views
Show that $e^z$ is continuous on $\mathbb{C}$
I know that $e^z$ is continuous on $\mathbb{R}$, but how would I show this rigorously on $\mathbb{C}$ using the $\epsilon - \delta$ definition of continuity?
I know how to begin:
If $|z - z_0| < \...
- 1,092
0
votes
3
answers
704
views
Proving $f(z)$ is continuous on whole complex plane
Question: define a function,
$$f(z)=\begin{cases}\frac{z \text{ Re(z)}}{|z|}&\text{ if $z≠0$}\\0, &\text{ if $z=0$}\end{cases}$$
Then prove or disprove $f(z)$ is continuous in entire ...
- 3,288
2
votes
1
answer
52
views
Complex Integral Function
I'm currently stuck on the following problem:
Let $\phi:[a,b]\times[c,d]\to\mathbb{C}$ be a continuous function and define $g:[c,d]\to\mathbb{C}$ by $$g(t)=\int_{a}^{b}\phi(s,t)\:ds.$$ Show that $...
- 1,888
2
votes
2
answers
3k
views
Proving complex function $f(z) = \frac1{1-z}$ is continuous on open disk
Prove that $f(z) = \frac1{1-z}$ is continuous on the open disk $\mathbb{D}_1(0) = \{z \in \mathbb{C}: |z|<1 \}$
I've been having a lot of trouble with this one. Here's my attempt thus far:
Fix $\...
- 4,844
3
votes
2
answers
5k
views
Show that the complex function f is continuous at the given point
Show that the function $f$ is continuous at the given point
$$f(z) =\begin{cases}
\dfrac{z^3 - 1}{z-1} & \text{if } |z| \neq 1 \\[6pt]
3 & \text{if } |z| = 1 \\
\end{cases}$$ $z_0 = 1$
I ...
1
vote
1
answer
356
views
The limit of $\frac{1}{z}$ in complex number
How to find $\lim_{z\to z_0}\frac{1}{z}$?
I want to make a $\delta-\epsilon$ argument to prove $\lim_{z\to z_0}\frac{1}{z}=\frac{1}{z_0}$.
Let $\epsilon>0$. There is a $\delta = \frac{1}{\...
- 1,425
1
vote
2
answers
567
views
Complex Analysis ( Limits at a point ).
We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables.
What I tried : ...
- 2,114
6
votes
1
answer
10k
views
Prove that $f(z)=z^2$ is continuous.
Prove that $f(z)=z^2$ is continuous for all complex and real values of $z$.
What I've got so far is:
Given $ \epsilon >0$ and $|z-z_0|<\delta$ after some calculations (which I've checked with ...
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