0
votes
1answer
19 views

At what point is this piecewise function continuous?

let $$f(z) = \begin{cases} z & |z|\leq 1 \\ |z|^{2} & |z|> 1 \end{cases}\ \text{where}\ z\in \mathbb{C}$$Does anyone could help me ?Thanks!
3
votes
0answers
27 views

Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
0
votes
0answers
20 views

product of the lengths of the segments $|AC|\bullet |BC|$ will be maximal.

Let there be a segment $AB$ the Diameter of the circle $S(0,1)$. Find all the points $C$ that belong to the closed circle $D^-(0,1)$ such that the product of the lengths of the segments $|AC|\bullet ...
0
votes
1answer
31 views

Show that $\Sigma_{n=1}^\infty |z_n| $ converges.

Assume $z_k = |z_k|e^{i\alpha_k}$ are complex numbers and that exists $0<\alpha<\pi/2$ s.t $\forall k -\alpha < \alpha_k<\alpha$ assume that $\Sigma_{n=1}^\infty z_n $ converges. we ...
1
vote
0answers
17 views

Show that $f'_n \rightarrow f'$ uniformly on every compact set. [duplicate]

Assume that $f_n,f:D(0,1)\rightarrow C$ are holomorphic. and $f_n \rightarrow f$ uniformly on every compact set. we need to show that $f'_n \rightarrow f'$ uniformly on every compact set. I kind of ...
0
votes
0answers
7 views

Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$

The question is: Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$ Now I've done some calculations and got that $T(z)=1/z$ but that is not enough..there are more and I'm not sure how ...
0
votes
0answers
15 views

f(iy)=1/(y-1) , what is the set of the points M(f(z))? [on hold]

when y changes in R - (1) , what is the set of the points M(f(z)) when z changes in iR We have f(iy)=1/ (y-1) , i have no ide what does this make , its not a circle , what is it
0
votes
0answers
32 views

Complex exponent problem

Find all numbers in complex plane that solves equation $e^z=4i$ Since $e^u e^{iv}=re^{i\Theta}$ it must be that $e^u=r \to u=\ln4$ and $v=\Theta+n2\pi \to v=\pi/2+n2\pi$. So the equation holds for ...
0
votes
0answers
27 views

Branches of complex logarithmic function

Attach analytic branch of some complex logarithm function and find range (image) of set if possible radius $\arg z =\pi /6$ positive $y$ axis circle $|z|=1$ ring $3 \leq |z| \leq5$ I'm not sure ...
0
votes
0answers
14 views

complex identity [duplicate]

I am trying to show that: $|e^{ix}-1|\le|x|$, x is a real-valued. First I tried Taylor series: ...
1
vote
1answer
25 views

Proving that an analytic function that maps on to {$z\in \mathbb{C}| |z-2|=1$} from some connected open set is constant

This is the approach I took to solve this but I got stuck. Suppose$f=u+iv\in $ {$z\in \mathbb{C}| |z-2|=1$} and that $f$ is analytic on an open connected set. Then we have that $(u-2)^2+v^2=1$. ...
0
votes
1answer
19 views

Radius of Convergence (Non-Series)

I am confronted with the following exercise: Compute the radius of convergence for the expansion at the point $z=4+4i$ for \begin{equation} f(z)=\frac{z^{5}e^{z}}{(2-z)(3i-z)} \end{equation} I ...
0
votes
1answer
34 views

The set of points in complex plane that satisfy a strict linear inequality is open

Let $S = \{(x,y)\in \mathbb C: y > 3x+2\}$. Show that $S$ is an open set. I can imagine what it looks like; a shaded region above a line. I also imagine that we must choose $ε$ (the radius ...
0
votes
2answers
43 views

How do I show $|\frac{i\overline{z}}{2} - \frac{i}{2}|=|z - 1|?$

I was looking over an example from our book concerning limits, and I'm having trouble seeing how this equality holds.
3
votes
2answers
73 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
0
votes
1answer
45 views

A problem on Complex differentiability

I following problem was given as a homework, I have explained how I approached it I need to know if it was correct and even then if it there wasn't any easier way, because that way only had tedious ...
1
vote
2answers
36 views

Prove $\left(z^n\right)' = nz^{n-1}$

I'm trying to solve this complex-variable problem: Prove, using direct Calculus, that $\left(z^n\right)' = nz^{n-1}$ ($n \in \mathbb{N}$). I tried the following steps to solve that: I saw that ...
0
votes
1answer
30 views

Express complex function in the form $u+iv$

One of the parts of the question I'm working on goes something like this: Express $z^i = \exp(i \log_I(z))$ in the form $u+iv$, where $u,v$ are real-valued functions, and the log is defined on the ...
0
votes
0answers
23 views

Analyze branch cuts and discontinuities of function $f(z)=\sqrt{1-z^2}$

Analyze the function $f(z)=\sqrt{1-z^2}$, where the square root is defined by the principal branch of the log function. Where does it have discontinuities? Here's what I did: We have $I = ...
0
votes
1answer
34 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
0
votes
0answers
42 views

Calculating the real and imaginary parts of a holomorphic function

Calculate the real and imaginary parts of the holomorphic function $f(z)=z^2\cos(z)-e^{z^3-z}$ and verify directly that each of these functions is harmonic. I believe I know how to the question, ...
0
votes
0answers
26 views

The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$ f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ I want to prove that this ...
0
votes
0answers
19 views

Complex number theory - limits

Can anyone please help me with those two limits? I have missed the first two weeks of a new semester, so really not experienced with this. $$a) \lim\limits_{z \to \infty} \frac{z^2-\overline z^2 + ...
0
votes
2answers
47 views

Question about a step in the proof of the Cauchy-Schwartz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwartz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le ...
1
vote
0answers
20 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
2
votes
1answer
37 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
0
votes
0answers
32 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
0
votes
1answer
19 views

Is this log identity true?

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?
1
vote
1answer
26 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
5
votes
5answers
58 views

Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
0
votes
3answers
28 views

All Values of a Complex expression

I am asked to find all values to $$\left(\frac{1-i}{\sqrt2}\right)^{1+i}$$ I do not know how to approach a power with complex part. Any help would be appreciated.
0
votes
0answers
12 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
0
votes
1answer
22 views

Straight Line Equation in Complex Plane

Hi there, I'm confused about the straight line equation in complex plane: how does "0 = Re((m+i)z + b)" come from "y = mx + b" ? I mean when I see "y = mx + b", I can draw a graph in my mind, but ...
3
votes
1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
0
votes
1answer
38 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
1
vote
0answers
63 views

Proving Ptolemy Theorem using complex number

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment: THE ...
0
votes
1answer
26 views

Prove that $|z||b-ad| \leq M $

I need to prove the following statement: $$ |z||\frac{az + b}{z+d}-a| <= M $$ with $a,b,c,z \in \mathbb{C}, |z| \geq 1 + |d|$ and $M\geq 0$. I have reduced this to $$ |z||b-ad| \leq M $$ Also $ad ...
1
vote
1answer
24 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
2
votes
1answer
24 views

Quadratic formula with complex coefficients

Let $a,b$ and $c$ be complex numbers. I'm trying to prove that this version of the usual quadratic formula: $$z=\frac{-b+(b^2-4ac)^{\frac{1}{2}}} {2a}$$ solves the quadratic equation ...
2
votes
0answers
54 views

If the sum of absolute values of complex numbers is at least $1$, then some subset of these numbers has absolute value at least $C$

There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem. Prove that there ...
3
votes
3answers
132 views

Find solution of equation $(z+1)^5=z^5$

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
4
votes
3answers
66 views

Why is Euler's formula defined for non-integer values?

Say that for some complex number $w$ $$e^{wi} = a$$ Now raise both sides to $1/4$. $$e^{wi/4} = a^{1/4}$$ Now $e^{wi/4}$ has a single defined value. Yet $a^{1/4}$ can have multiple values. So why ...
4
votes
1answer
79 views

Why is Euler's formula valid for all $n$ but not De Moivre's formula?

The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values. It then goes on to say that this ...
0
votes
0answers
29 views

Guidance for complex numbers/analysis problem needed [duplicate]

I'm looking at this one problem in a book of mine, but I can't even seem to start it. Let $z_1,z_2,...$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every ...
2
votes
3answers
127 views

Proving $\arg(zw)=\arg(z)+\arg(w)$

This is my attempt I know this is incomplete or may even be wrong. Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient ...
2
votes
1answer
41 views

Find loci of the points in complex plane such that $\mathrm{Im}(\frac{z-z_1}{z-z_2})^n=0$

Find loci of the points in complex plane such that $$\mathrm{Im} \left (\frac{z-z_1}{z-z_2}\right )^n=0,$$ where $n\in\mathbb{N}$, $z_1, z_2$ are the given points in $\mathbb{C}$. When $n=1$, it ...
1
vote
3answers
81 views

Every line or circle in $\mathbb{C}$ are solution sets to the equation…

Here is a complex analysis homework problem I can't quite figure out: Prove that every line or circle in $\mathbb{C}$ is the solution set of an equation of the form $a|z|^2+\bar{w}z+w\bar{z}+b=0$, ...
0
votes
0answers
42 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
2
votes
2answers
38 views

Complex analysis: Rewrite $\cos^{-1}{i}$ in algebraic form

I'm stuck in this problem (complex analysis), my answer is not the one reported in the book: Rewrite $\cos^{-1}{i}$ in the algebraic form. A: $k\pi + i \frac{\ln{2}}{2}\ \forall\ k \in \mathbb{Z}$ ...
1
vote
1answer
44 views

Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...