Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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1answer
18 views

Schwarz Lemma, an onto map with $f'(0)>0$ is the identity

Let $f$ be $1-1$ holomorphic on unit disk onto itself. It satisfies (a) $f(0)=0$, (b) $f'(0)>0$. We need to prove that $f(z)$ is equal to $z$. I am stuck here, because I can prove using Shcwarz ...
0
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1answer
34 views

Unique properties of pure Imaginary numbers?

Are there any non trivial properties unique of the imaginary numbers? By trivial I mean stuff like $\bar a=-a$.
-2
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1answer
46 views

The real part of the sum $(i-1)+(i-1)^2+(i-1)^3…+(i-1)^{2013}$?

I'm not sure how to go around this one. Factorizing doesn't seem to work and there isn't a clear pattern to work by that I see.
0
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1answer
18 views

Prove that if $Re(z)>0$ then $|z+\sqrt{z^2-1}| \ge 1$

This is probably a very basic question in complex numbers. First define $\sqrt{w} := \sqrt{|w|}e^{i(Arg(w)/2)}$ where Arg is the principal argument function. Prove that if $Re(z)>0$ then ...
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0answers
23 views

Complex Geometry Problem

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find [$PA_1^2 + PA_2^2 + \dots ...
2
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3answers
215 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
5
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4answers
59 views

Find all solutions to the following equation: $x^3=-8i$

Find all solutions to the following equation: $$x^3=-8i$$ I found the modulus, $$r=8$$ $$\operatorname{arg}(x)=\arctan(-8/0)=-π/2+2πk$$ By De Moivre's Theorem: ...
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0answers
17 views

Algorithm to find value where complex numbers meet on the unit circle. [on hold]

I'm trying to find the value at which 4 points are meeting on the unit circle. These points are eigenvalues of the translation operator $T$. By varying $\lambda$ the eigenvalues change. Background: ...
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1answer
31 views

complex numbers- how do I prove the following statement? [duplicate]

given: $$ \left | z_{1}\right |=\left | z_{2}\right |=...=\left | z_{n}\right |=1 $$ How do I prove: $$ ...
1
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1answer
17 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
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0answers
29 views

Does rule of three make sense for other than real numbers?

I'm currently working on a software tool which can make calculations based on the rule of three. I can make it more simple and just support real numbers, or I can make it "universal" to support ...
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5answers
37 views

Complex Numbers Question, IIT JEE [2006]. Please tell me whether I solved it properly?

$Q.$The value of $\sum\limits_{k=1}^{10}(\sin{\frac{2k\pi}{11}-i\cos\frac{2k\pi}{11}})$ is-? I solved it like this- $\frac{\sum\limits_{k=1}^{10}(\cos{\frac{2k\pi}{11}+i\sin\frac{2k\pi}{11}})}{i}$ If ...
0
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0answers
29 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the real matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
2
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1answer
39 views

Area of triangle formed by angle bisector, altitude and median

Question:- Given a triangle ABC with side length a, b and c. Calculate the area of a triangle in terms of a, b and c formed by angle bisector from vertex A, altitude from vertex B and median from ...
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2answers
22 views

Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
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3answers
34 views

How is this step done? $\left|\frac{i\overline{z}}{2} -\frac i2\right|=\frac{|z-1|}{2}$

Absolutely everything makes sense other than what is in red. How is this step completed? Let us show that if $f(z)=\dfrac{i\overline{z}}{2}$ in the open disk $|z|\lt 1$, then$$\lim ...
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1answer
25 views

Finding eigenvalue and eigenvectors of a matrix containing an imaginary number

How do you solve for the eigenvalues given the matrix? \begin{matrix} i & -2 \\ 1 & 0 \\ \end{matrix} I know how to get the characteristic polynomial Ca(X); X^2 - ...
3
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2answers
77 views

how can I simplify this $\sqrt{i}+\sqrt{2i}+\sqrt{3i}$

Is there an easy way to simplify the $$\sqrt{i}+\sqrt{2i}+\sqrt{3i}$$
0
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1answer
43 views

Cube root of $\omega$

What is the cube root of $\omega$? where $\omega$ is the non-real cube root of $1$. I got the cube root of $\omega$ in terms of $\omega$! I took cube root of $\omega$ as $x+iy$ then cubed it. I got ...
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3answers
58 views

Find all values of $\sqrt[4]{-1+i}$

Okay. I know how to solve for all values of $\sqrt{-1} $ but $\sqrt{-1+\iota} $ confuses me a bit. I got the value of r to be $\sqrt 2 $ I ended up with this: $ z_k = ...
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4answers
68 views

is 1 greater than i?

I'm not sure this question even makes sense because complex numbers are a plane instead of a line. The magnitudes are obviously the same because i is a unit vector, but is there any inequality you can ...
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2answers
22 views

$Im(\cosh z)=\sinh x\sinh y$ and $|\sinh z|^2=\sinh^2(z)+\cosh^2(z)$

I have problems in two issues of complex variables ... 1) Prove that $Im(\cosh z)=\sinh x\cosh y$, if $z=x+iy$. I tried to expand $\cosh z= ...
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1answer
57 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
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6answers
65 views

Find the three roots of $z^3 = -i$ in the form $a+ib$. [on hold]

Find the three roots of: $$z^3 = -i $$ in the form $a+ib$.
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0answers
24 views

Locus of complex numbers.

Question Let $P(x,y)$ be the point on an Argand diagram representing the complex number $u=x+iy$ and satisfying the equation \begin{align*} \vert u \vert=k\vert u+a\vert, \end{align*} where $k$ is ...
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1answer
35 views

$|z^i|<e^\pi,\;\;\forall z\in\mathbb{C}-\{0\}$

Good morning people ... Do you have any idea to help me prove that $$|z^i|<e^{\pi}$$ for $z\in\mathbb{C}-\{0\}$. I tried to do $z^i=e^{i\ln z}=e^{i(\ln r+i\theta+2k\pi i)}$ if $z=re^{i\theta}$, ...
2
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0answers
35 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...
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1answer
19 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
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1answer
27 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
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1answer
19 views

Cartesian equation of the loci of 'z', [on hold]

Can someone FIND |z-3j|+|z+7j|=12
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1answer
31 views

Consider the equation $|z + 3i|=3|z|$ for complex z and give a geometric description of the set S of all solutions.

Writing $z$ in the form $a+ib$ and then rearranging gives $-8a^2-8b^2+6b+9=0$. The most promising form I could manage from this is $(b-\frac{3}{8})^2=(\frac{9}{8}-a)(\frac{9}{8}+a)$ but I still do not ...
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1answer
52 views

Can someone find z^10 quickly, given z? [on hold]

z = 1 + i z^10 = ? I need a solve of this quick please !
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1answer
28 views

How can I split this into its' real and imaginary parts, and simplify?

Essentially, I want to prove that $| \sum_{k=1}^n e^{ik}|$ is bounded. If I obtain an expression for this sum: $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ I am not sure how to proceed ...
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1answer
42 views

Help with simplifying this - where have I made an error

$$e^i \frac{e^{in}-1}{e^i-1}$$ $$e^i \frac{e^{in}-1}{e^i-1} \cdot \frac{e^{-i} -1}{e^{-i} -1}$$ $$e^i \frac{(e^{in}-1)(e^{-i}-1)}{(e^{-i}-1)(e^i-1)}$$ $$e^i ...
3
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5answers
43 views

Find the root of a complex number

Find all complex numbers $z$ such that $$z^2=12−16i,$$ and give your answer in the form $a+bi$. We set $$z= a+bi,$$ thus, $$z^2 = (a^2 - b^2) + (2ab)i.$$ Equating both $z^2$ we have $$ a^2 ...
2
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1answer
30 views

How to solve a Complex equation involving the conjugate: $(x+iy)^2-2(x-iy)+1=0$

I want to find a Complex value for $z$ that satisfy the equation: $$z^2-2z^*+1=0$$ But i have never seen the conjugate taking part of an equation. What i have tried is give $z$ some components ...
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2answers
22 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
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1answer
27 views

Solve z in an expression involving complex conjugates.

Solve for z, and give your answer in the form a+bi. $$\overline{z+2-2i} = {2z + 5 - 7i}$$ I know fully understand the concept of complex numbers and complex conjugates. I've found that the answer is ...
0
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2answers
43 views

Understanding complex numbers

I need to show that $$\left | \sum_{k=1}^n e^{ik}\right | $$ is bounded Now I am given that $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ But have little idea of how to proceed further and ...
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3answers
44 views

Roots of Unity, Precalculus

(a) Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 ...
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0answers
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Prove that if $g = r +ip$ is analytic on $C$ and $r(x,y) \leq M$, with $M > 0$, for all $(x,y)\in C$, $g$ is constant.

Let $g = r +ip$ be analytic on $C$. If for some $M > 0$ we have $r(x,y) \leq M$ for all of $C$, then $g$ is constant. The theorem is given without proof in my notes and I can't find any examples ...
0
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1answer
21 views

Shade on your Argand diagram the region $\frac{\pi}{4}\,{\le}\,\arg\,z\,\le\frac{\pi}{2}$

Is this saying the region from $\arg\,z=\frac{\pi}{4}$ to $\arg\,z=\frac{\pi}{2}$ in an anticlockwise direction? How would you represent the region from $\arg\,z=\frac{\pi}{4}$ to ...
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0answers
23 views

Locally Lipschitz complex function [on hold]

I want to study the property of being locally Lipschitz for the following function $$f(z)=\vert z\vert^\gamma z^2$$ with $\gamma\in\mathbb{R}$. Some hints to study this problem?
1
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1answer
35 views
+50

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
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0answers
33 views

How to integrate complex numbers? [on hold]

Complex numbers have 2 variable so does it's integration entail contour integration or can we integrate assuming one variable to be a constant in terms of the other or do we try to find a relation ...
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0answers
34 views

Complex Analysis Exam tomorrow, what are some good to know facts? [on hold]

The course covers differentiation, integration, series, and a lot of theorems. What would you say is crucial to know for an exam of undergrad complex analysis?
2
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4answers
56 views

Showing that linear transformations $1, T, T^2, T^3 ,\dots $ do not span the set of linear transformations of $ \mathbb C^n$ into $ \mathbb C^n$

Suppose $T : \mathbb C^n \rightarrow \mathbb C^n$, $n \geq 2$ is a linear transformation. Show that the linear transformations $1,T, T^2, \dots$ do NOT span $L(\mathbb C^n, \mathbb C^n)$, the set of ...
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2answers
31 views

Trisecting a line in the complex plane

We have $x = 11-13i$ and $y = 35-i$. $a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$. I'm not sure where to start. Would a ...
0
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0answers
20 views

Linear interpolation of complex numbers

Does it make sense to say that, if I have two numbers $X = a + bi$ and $Y = c + di$, I can approximate a point between them as $Z = \frac{a + c}2 + \frac{b + d}2i$, interpolating the real and complex ...
0
votes
2answers
39 views

Minimising $|a+bw+cw^2|$ such that a,b,c are consecutive integers?

Suppose we are given a expression $k=|a+bw+cw^2|$ such that $w$ is cube root of unity ($w\neq1$) such that $\{a,b,c\}$ are consecutive integers , then how can we minimise value of expression ? I was ...