All Questions
34
questions
0
votes
1
answer
57
views
In complex number system, sin z and cos z are unbounded and periodic. But they are continuous also. How can that be possible?
I know that a continuous periodic function must be bounded because if a function is continuous and periodic, its graph will have to turn at certain points to reattain the values and hence, it cannot ...
- 13
5
votes
6
answers
628
views
What is $\sqrt{-1}$? circular reasoning defining $i$.
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \...
- 6,620
0
votes
2
answers
128
views
As a matter of fact, it is impossible to find a continuous $f$ such that $(f(z))^2=z$ for all $z$. ("Calculus Fourth Edition" by Michael Spivak.)
As a matter of fact, it is impossible to find a continuous $f$ such
that $(f(z))^2=z$ for all $z$. In fact, it is even impossible for
$f(z)$ to be defined for all $z$ with $|z|=1$.
To prove this by ...
- 1,138
0
votes
1
answer
66
views
Prove that $f(z)=\int_0 ^1 t^z dt$ is continuous
Let $$f(z)=\int_0 ^1 t^z dt.$$ Prove that $f$ is holomorphic on $\{\Re(z)>-1\}$.
My attempt: First notice that $$|t^z|=|e^{z\log(t)}|=e^{\Re(z\log(t))}=e^{\log(t)\Re(z)}=t^{\Re(z)},$$ and thus $$\...
- 146
2
votes
0
answers
50
views
Surface integral of a complex Log function
I am trying to calculate the surface integral of a complex Log function i.e.
$$ \int\int_{|z|<1}{Log(x+i y-(x_0+iy_0)))dxdy}$$
where $z=x+iy$ and $x_0,y_0 \in \mathbb{R}$ .
I know that for analytic ...
- 21
2
votes
1
answer
218
views
Continuity of $\text{Im}\frac{z}{z-1},\frac{\text{Re }z}{z},\text{Re }z^2,\frac{z\text{Re }z}{\left|z\right|}$
Find the set of points for which the given functions are continuous on that points.
$\text{Im}\frac{z}{z-1}$
$\frac{\text{Re }z}{z}$
$\text{Re }z^2$
$\frac{z\text{Re }z}{\left|z\right|}$
My ...
- 69
0
votes
1
answer
59
views
Continuity of a complex variable function.
Let $$ 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
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
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
1
vote
1
answer
29
views
Continuity of Complex Function of Two Functions Dependent of Re(z)
I believe this function is continuous at $0$ and hence is a continuous function.
It's continuous if $\lim_{z \rightarrow 0^-}\bar{z} = 0$, is that right?
If so, how do I prove that that's the limit? ...
- 1,250
1
vote
2
answers
118
views
If $f$ is a mapping from $\overline{B_1(0)}$ to itself that is continuous and analytic, prove that $f$ has a fixed point.
I am new to this but I kept on looking answer about this problem
Suppose $\displaystyle{f: \overline{B_1(0)} \rightarrow \overline{B_1(0)}}$ is continuous and $f$ is analytic in $\displaystyle{B_1(0)}$...
1
vote
2
answers
96
views
Show there exists $z \in \mathbb{U}$ such that $\prod_{k=1}^n (z - a_k) \in \mathbb{U}$
The exercise
Let $n\geq 3$ and $a_1, \dots, a_n \in \mathbb{U}$. Show there exists $z \in \mathbb{U}$ such that $\prod_{k=1}^n (z - a_k) \in \mathbb{U}$.
My try
Writing $z = e^{i \theta}$ and $a_k = e^...
- 1,214
1
vote
2
answers
634
views
Why is the argument function defined in Complex Analysis discontinuous?
I've been working through Spivak's Calculus and I'm currently working through Ch. 26 on Complex Functions. One of the functions they define is the argument function. Recall that the argument of a ...
- 5,659
1
vote
1
answer
69
views
How to prove that the n-th root is not continuous using the Fundamental Group
I need to prove that the function $g:\mathbb{C}^∗ \rightarrow \mathbb{C}^*$ such that $(g(z))^n=z$ is not continuous using the fundamental group.
I tried to use the argument in the question: How to ...
- 25
1
vote
1
answer
199
views
How to prove that the complex logarithm is not continuous using the Fundamental Group
I need to prove that the function $f:\mathbb{C^*} \rightarrow \mathbb{C} ; \exp{(f(z))} = z$ is not continuous using the fundamental group.
I´ve found this Does every continuous map induce a ...
- 113