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
16 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
1answer
29 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
14 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
33 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
24 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
16 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
22 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
53 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
26 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
15 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
49 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
23 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
52 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
130 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
77 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
124 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
38 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
76 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
37 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
42 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 ...
0
votes
2answers
32 views

Simplify $w=\frac{(1+i)z-i+1}{iz-1}$

I have difficulties understanding how this expression $$w=\frac{(1+i)z-i+1}{iz-1}$$ is simplified to this $$w=1-i-2\cdot\frac{1+i}{z+i}$$ Here are some steps from my exercise notebook: ...
0
votes
0answers
12 views

Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $ \gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
0
votes
1answer
22 views

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$ I have concluded that $xu_x = yu_y$. Not sure how to proceed ...
0
votes
0answers
34 views

$√2|z|≥|Rez|+|Imz|$ [duplicate]

I just came across this simple inequality which I am finding it difficult to prove. For any complex number z I need to prove that the following inequality holds $$√2|z|≥|Rez|+|Imz|$$ I would much ...
3
votes
1answer
54 views

Suppose that $a_0 >a_1 >…>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ [duplicate]

Suppose that $a_0 >a_1 >...>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ Not sure where to begin with this. Any suggestions? Thanks.
1
vote
2answers
41 views

Finding $\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$

$\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ I am having trouble getting a final answer that makes sense to me. Here is what I tried: ...
0
votes
0answers
29 views

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$ By Picard we have that every number except maybe one is, but that is all I've got. Help would be great! ...
6
votes
4answers
175 views

Can I conjugate a complex number : $\sqrt{a+ib}$?

Can I conjugate a complex number: $\sqrt{a+ib}$ ? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ because ...
0
votes
0answers
46 views

Modulus of a complex function

Suppose $\alpha$ and $\beta$ are two arbitrary complex numbers. Let $$f_{\pm}(\alpha, \beta)=\frac{\frac{-\alpha}{\alpha+\beta}\pm ...
1
vote
3answers
35 views

A problem on modulus of complex numbers

Let $ z,a \in \Bbb C $ and suppose that $ |a| <1$ Prove that $ |z-a| \leq |1- {\bar a} z| $ iff $|z| \leq 1 $ I couldn't find a way to approach the sum without using $a,z$ and $x+iy$, from which ...
0
votes
1answer
29 views

A circular path connecting two complex numbers

For any two complex numbers $z_1$ and $z_2$, $f(t)$= $z_1+t(z_2-z_1) $is a path in $ℂ$ where $t∊[0,1]$. The image of this path is a line segment. Is there a way of getting a similar path but to ...
1
vote
1answer
35 views

Prove that if $z_n \rightarrow z$ then $\theta_n \rightarrow \theta$ and $r_n \rightarrow r$.

Suppose that $z_n,z \in G = \mathbb{C} - \{z:z\leq 0\}$ and $z_n=r_ne^{i\theta_n}, z = re^{i\theta}$ where $- \pi < \theta_n,\theta< \pi$. Prove that if $z_n \rightarrow z$ then $\theta_n ...
0
votes
2answers
21 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
0
votes
2answers
52 views

geometric description of set of complex number

A set of complex number: $$S=\{ z\in \Bbb C : |z|=\lambda |z-1|\}$$ what's the geometric description? I try to draw it ... which seems like a circle but cannot find the equation to describe it..
2
votes
3answers
32 views

Factorization of $z^4 +1 = (z^2 - \sqrt 2z+1)(z^2 + \sqrt 2 z+1)$ for complex z

How can I get this equation from LHS to RHS by using the four roots of $z^4 +1 = 0$ are $z=\pm\sqrt{\pm i}$ $$z^4 +1 = (z^2 - \sqrt2 z+1)(z^2 + \sqrt2 z+1)$$
0
votes
1answer
26 views

Supremum of the set $\{\operatorname{Re}(iz^3+1) : |z|<2\}$

I need to find supremum of the set of all real numbers of the form $\operatorname{Re}(iz^3+1)$ such that $|z|<2$. By the inequality $-|w|\le \operatorname{Re}(w)\le |w|$ we have ...
1
vote
2answers
38 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
1
vote
1answer
34 views

Complex roots (review) (advise)

I have to find the complex roots and want a review of my procedure to see if is correct A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$ ...
0
votes
1answer
33 views

the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
-1
votes
2answers
76 views

Can I conjugate a complex number: $\sqrt{a+ib}$?

Can I find the conjugate of the complex number: $\sqrt{a+ib}$? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ ...
1
vote
0answers
35 views

Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...