Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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2answers
51 views

Automorphism of unit disk without zero

Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. $g(z)= e^{i{\...
2
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2answers
48 views

How to sketch the region on the complex plane? [duplicate]

I am going through a basic course on complex analysis. I have a problem in understanding the following. E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...
2
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1answer
72 views

How to simplify $\sqrt{-8}$

How would I go about simplifying square root of $-8$? I know I can rewrite that as $\sqrt{(-1)(8)}$, and then I would get $i\sqrt{8}$, but how do I simplify that $8$ further? Thanks for your help.
1
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1answer
38 views

graphing a circle in the complex plane? [closed]

The ellipse seemed rather simple: Defining the equation of an ellipse in the complex plane But Wolfram won't graph it with equal axes. http://www.wolframalpha.com/input/?i=abs{%28x%2Biy%29}%2Babs{%...
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3answers
38 views

How are the following factors 'linear'.

What does it mean for factors to be linear? Q: Find the four linear factors of: $$z^4+z^3+z^2+z+1$$ I got the following: $$(z-e^{i \pm {2\pi \over 5}} )(z-e^{i \pm {4\pi\over 5}} )$$ I though ...
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1answer
67 views

What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $ e^{i\theta} $ versus $ e^{a+bi} $? Am I correct in saying that these give the same information but offer convenience in different situations? ...
0
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3answers
58 views

How to derive values for $i$ raised to negative integers? [closed]

This link states that the values of $i$ raised to the power of negative integers. How can we derive these values from the positive powers?
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6answers
99 views

Homework quesiton: Find $p$ and $q$ so that $(p+qi)^2=3-4i.$

I got as far as taking the square root of both sides, and I'm ashamed to say that I'm already stuck. Any pointers? In regards to the comment, I got as far as $q^4+3q^2-4=0$ by equating the parts that ...
2
votes
2answers
47 views

Taking Mod on both sides, mathematically correct?

When given a equation containing complex numbers such as $$ \frac{a+ib}{c+id} = x + iy$$ and required to prove $$ \frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$ Is taking the mod of both sides a legal ...
1
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3answers
90 views

Proof Involving Imaginary Number: Where's the wrong one? [duplicate]

Here are the propositions: $$i=\sqrt{-1}$$ $$i^2=-1$$ $$(i)(i)=-1$$ $$\sqrt{-1}\sqrt{-1}=-1$$ $$\sqrt{(-1)(-1)}=-1$$ $$\sqrt{1}=-1$$ There's an error in the propositions above. I think it's in the ...
12
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4answers
1k views

Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
0
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3answers
37 views

modulus and argument of $(-4\sqrt{3}-4i)^3$?

Any fast method to obtain the modulus and argument of $(-4\sqrt{3}-4i)^3$? If i use the exponential form to solve it, is it good?
1
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1answer
40 views

minimum of $F(z) = 4\left|z-1-2i\right|^2+13\left|z-4-5i\right|^2+12\left|z-2-7i\right|^2+23\left|z-6i\right|^2$

Find Minimum value of $F(z) = 4\left|z-1-2i\right|^2+13\left|z-4-5i\right|^2+12\left|z-2-7i\right|^2+23\left|z-6i\right|^2$ $\bf{My\; Try::}$ Put $z=x+iy\;,$ We get $$f(x,y) = 4(x-1)^2+4(y-2)^2+...
3
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1answer
109 views

Finding all $z\in \mathbb{C}$ such that the series $\sum\limits_{n=1}^{\infty} \frac{1}{1+z^n}$ converges

I am trying to find out all $z\in \mathbb{C}$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+z^n}$ converges. I notice that for $\left|z\right|\leq 1$, we have $\left|1+z^n\right|...
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1answer
18 views

Can you help me to understand this Magnitude/Phase to Real/Imaginary conversion?

I've a module/function that takes an array of magnitudes/phases and get to me the real/img results. These are the input values. Magnitude values: ...
0
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2answers
47 views

A sum of powers of primitive roots of unity

For the primitive roots of unity $\omega_n = e^{i2\pi/n}$ I want to prove that $$\sum_{k=0}^{n-1} \omega_n^{lk} = 0$$ if $n$ doesn't divide $l$. I have already proven the well-known result $$\sum_{k=...
1
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2answers
41 views

Help with De Moivre's Theorem: Complex Numbers

I have a homework problem which goes: Given $z^n=(z+i)^n$, using de Moivre's Theorem, show that $z=\frac{i}{e^\frac{i2k\pi}{n}-1}$ What steps should I take in tackling this question? It's a 2 mark ...
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2answers
41 views

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$
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1answer
71 views

Image drawing complex analysis [closed]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
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1answer
51 views

Roots of unity whose sum and product are known

Is cube root of unity is a complex number I know the sum is 0 and product is -1 but I am somewhat confused please give me some idea. Thanks in advance
2
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0answers
41 views

sum of subset of complex numbers [duplicate]

let there be $\{z_1 ,..., z_n\}$ a group of complex numbers. Show that there's a subset $J \subset \{1,...n\}$ so that $$\lvert \sum_{k \in J}z_k \rvert \ge \frac{1}{4\sqrt2}\sum_{i=1}^n\lvert z_i \...
2
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0answers
31 views

Find the best constant $C_{n}$ such this complex inequality

nd we can consider this problem In general? if $|z_{1}|=|z_{2}|=\cdots=|z_{n}|=1$ if there exist complex $z(|z|=1)$ such $$\sum_{i=1}^{n}\dfrac{1}{||z-z_{i}||^2}\le C_{n}$$ find the best $C_{n}$? $$...
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0answers
10 views

Where does the formula $\zeta_{qj} z=\frac{1}{2}H_{qj} z+\frac{1}{2}h_{qj}$ come from?

I am reading the book space filling curves by Hans Sagan. I am reading about the complex representation of the Hilbert Curve. I came across the formula $\zeta_{qj} z=\frac{1}{2}H_{qj} z+\frac{1}{2}...
1
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2answers
22 views

Number of roots of a complex exponent

There are $p$ solutions to $\sqrt[\frac{p}q]1$, if $\frac{p}q$ is a fraction in lowest terms. I have found on this website that an irrational exponent has infinite roots. But what about $\sqrt[a+bi]1$...
0
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1answer
47 views

How to find the General expression of $\sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k}$ [duplicate]

Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity $$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$ Need help ...
1
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3answers
75 views

$e^{a 2\pi i} = (e^{2\pi i})^a$.

When $a$ is any real number , Is it true $e^{a 2\pi i} = (e^{2\pi i})^a$ ? The reason why I ask this question is that I met this situation wheter this equality hold in Calculating Integral in Complex ...
0
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2answers
58 views

Complex numbers inside determinant

Let $ \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
1
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1answer
173 views

Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
0
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2answers
17 views

Complex Conjugate roots with non real coefficients

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ...
2
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1answer
36 views

Visualizing a complex function

Ever since I learned about complex valued functions I've been wondering if there was a better visualization for them. Obviously we can't visualize four dimensions, but I was wondering if it would be ...
0
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3answers
78 views

Solve in $\mathbb{C}$ : $|z-i| = |z-1|$

I just had that question in my final exam Solve in $\mathbb{C}$ : $|z-i| = |z-1|$ and I couldn't do it. I found a similar thread here : Showing that $\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is an open ...
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0answers
27 views

Bounded real parts of the solutions of an equation

I'd like to show that, with $a,b>0$, the real parts of the solutions $z_n$ of the equation $$ az+\sqrt{z^2-ib}\tanh\sqrt{z^2-ib}=0 $$ are bounded. An indication for that can be found if we ...
0
votes
1answer
29 views

biholomorphic on unit disk

Let $D$ be the unit disk and $f: D\rightarrow G$, $\; p_1$ the maximum value of $dist(f(z),f(0))\;$ and $p_2$ the minimum value of $dist(f(z),f(0))$ for $z\in \partial \bar G$ Prove that : $|f(z)-...
1
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1answer
57 views

Evaluate $\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}$

Evaluate $$\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}.$$ The first thing i noticed was that $$\cos \frac{\pi}{7}=\frac{\zeta_{14}+\zeta_{14}^{-1}}{2},$$ where $\zeta_{14}=e^{2\pi i/14}$...
0
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1answer
31 views

Sine and cosine solutions of a differential equation

I have to solve a differential equation with constant coefficient such as$$ay'''+by''+cy'+dy=f(x)$$ which has for a characteristic equation$$P_c(\lambda)=a\lambda^3+b\lambda^2+c\lambda+d=0$$First I ...
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0answers
20 views

Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
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2answers
46 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
2
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1answer
43 views

Least value of complex expression

If $z_{1},z_{2},z_{3},z_{4}$ are $4$ points on a circle $|z| = 1$ such that $z_{1}+z_{2}+z_{3}+z_{4}=0\;,$ Then least value of expression $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{4}|^2+|z_{4}-...
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1answer
94 views

Why is De Moivre's theorem not generalised for $(\sin x+i\cos x)$?

A representation of the form $(\sin x+i\cos x)^n$ can be reduced as follows $$( \sin x + i \cos x )^n$$ $$( \cos (90-x) + i \sin(90-x) )^n$$ $$( \cos (90n - nx) + i \sin(90n - nx) )$$ Now for all ...
2
votes
4answers
66 views

Linear algebra : Solving $i \cdot\bar{z} = 2 +2i$

$i\cdot\bar{z} = 2+2i$ I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$ Then $ai+b=2+2i$ (because $i^2=-1$) When 2 complex numbers are equal you usually can equal their parts Ex: $2+2i=a+bi$...
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1answer
38 views

How does analytic continuation lets us extend functions to the complex plane?

I'm trying to understand analytic continuation and I noticed on wolfram that it allows the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic ...
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0answers
47 views

Write complex number $z^5$ in the form $r(\cos θ + i\sin θ)$

Let $z = 4(\cos \frac 8 7 π + i\sin\frac 8 7π)$ be a complex number. Find $z^5$ in the form $r(\cos θ + i\sin θ)$ with r being a positive real number, and with $0 ≤ θ < 2π$. My attempt: $z^5 = ...
1
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1answer
50 views

The identity $ \sqrt[n]{z}\sqrt[n]{w} = \sqrt[n]{zw}$ for complex numbers

In the general case, when $z$ and $w$ are two complex numbers, we have that $ (1) \sqrt[n]{z}\sqrt[n]{w} \neq \sqrt[n]{zw}$ For example, $\sqrt{-1}\sqrt{-1} \neq \sqrt{-1.-1} = 1$. However, there ...
1
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1answer
24 views

Holomorphic functions and complex conjugation

Suppose I have given two holomorphic functions $g,f:\mathbb{C}\backslash(-\infty,1]\rightarrow \mathbb{C}$ and I know that $\overline{ g(z)}=f(z)$ for all $\vert 2-z \vert <1.$ I am wondering if ...
0
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0answers
23 views

discrete logarithm with complex numbers

let $z = a + bi$ where $a,b$ are integers on $[0,N)$ let $a + bi \mod t = (a \mod t) + (b \mod t) \cdot i$ Consider the problem of finding $e$ where $z^e \mod N = c$ and $c, N$ and $z$ are known. Is ...
0
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1answer
17 views

inequality by taking reciprocal or other way to check if pole lies inside unit circle

If $ a^2$ <1 is given in the problem then how do we prove that the pole z=1/a lies outside the unit circle ?
0
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1answer
42 views

Sufficient condition on open subsets to be equal

Let $U,V\subseteq\Bbb C$ connected open non empty, such that their closure in $\Bbb C$, say $\overline U,\overline V$, be simply connected. Then, is it true that, if $$ U\cap V\neq\emptyset\\ \...
2
votes
2answers
121 views

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number ...
0
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0answers
14 views

Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
0
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1answer
18 views

Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$.

I'm starting to see complex numbers in algebra. I've missed a few classes and I have exercises similar to this one: Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$. ...