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

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1answer
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Finding complex roots of integer polynomials

How would one find approximates for complex root of polynomial with integer coefficients,I know for example the Newton's method $$x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$$ Anyway is it possible to ...
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2answers
54 views

Failure of De Moivre's Theorem

I know that De Moivre's Theorem does not necessarily work for non-integer powers. The classic counter-example is by considering $\left (\cos \theta + i \sin \theta \right )^n=\cos n\theta + i \sin n ...
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2answers
81 views

Euler's Formula, Square root [on hold]

I have been doing some work on Euler's formula. I need to come up with the formula for $e^{i\theta}$ where $i = \sqrt{-1}$. I know $e^{i\theta} = \cos \theta + i\sin \theta$, but I am not sure how ...
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2answers
35 views

Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
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1answer
24 views

Calculating $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$.

I need to calculate $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$ and $r \in \mathbb R$. My Attempt: $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}=\sum_{n=0}^\infty r ...
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4answers
50 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
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1answer
39 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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1answer
28 views

Equality of complex numbers

I'm currently reading some notes on Complex Numbers and came across this 'proof' regarding the equality of complex numbers. Claim: Two complex numbers $a+bi$ and $c+di$ are equal iff $a=b$ and $c=d$, ...
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2answers
28 views

rationalize the complex number multiplication rule

For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of $(a,b)$ and ...
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1answer
23 views

Complex numbers - locii

I have been asked to solve the following and represent the answer graphically: A) $| \arg z - (\pi/4) | < (\pi/2)$ I understand that this means the difference between the argument of $z$ and ...
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2answers
25 views

Is ring of Gaussian rationals in unique factorization domain?

Instead of Gaussian integers, let us think about Gaussian rationals, where $a$ and $b$ in $a+bi$ are rational numbers. Then would ring of Gaussian rationals be in unique factorization domain?
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1answer
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square of complex numbers

I have this equation from here: but it is not equal to: $$(a + bi)^2 = a^2 + 2abi + (bi)^2.$$ could someone explain me what is the difference between this two calcultion?
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0answers
62 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
2
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1answer
44 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...
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3answers
66 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
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1answer
30 views

Arg(z) from $z^n$

$z \in \mathbb{C}$. If the principal argument of $z^n$ is in the quadrant $q$, what is the complete set of values for $Arg(z)$? For example if $n = 3$ and $q = 2$, how could I find all values of ...
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1answer
33 views

Equilateral triangle from complex numbers

We know that $z_1+z_2+z_3=0$ and $|z_1|=|z_2|=|z_3|=1$ where $z_1,z_2,z_3$ are complex numbers. How can we show that the images of $w_1=z_1^4*z_2^3 , w_2=z_2^4*z_3^3 , w_3=z_3^4*z_1^3$ form an ...
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1answer
52 views

Polar Coordinate usind De Moivre’s Theorem

I need the solutions for the following problem: Find all solutions over $\mathbb{C}$ to the equation $x^3=i^2$. I tried using De Moivre’s Theorem can't get around it. Note: The question originally ...
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1answer
211 views

Interpret to a complex plane!

$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}$The question is: Interpret $$ \Re z + \Im z = 1 $$ geometrically in the complex plane. Writing $z = x + yi$, the condition ...
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2answers
52 views

Another way to solve this problem with complex expressions

The problem is this: Express $x$ and $y$ with $u$ and $v$, if $\dfrac{1}{x+iy} + \dfrac{1}{u+iv} = 1$ Where $x,y,u,v \in \mathbb{R}$, and $i^2 = -1$. I could solve it, but I used a hairy and ...
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2answers
222 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
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3answers
100 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
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1answer
74 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
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2answers
77 views

Find all complex number $z\in\Bbb{C}$ such that $\vert z\vert=\vert z^{-1}\vert=\vert z-1\vert$

Find all complex number $z\in\Bbb{C}$ such that $$\vert z\vert=\vert z^{-1}\vert=\vert z-1\vert$$ I tried to write $z=a+ib$, clearly $z=1$ is not a solution. I have to solve $$\left\{ ...
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2answers
49 views

Let $z = 2 + 2i$ Find all complex numbers $w$ such that $w^4 = z$ [on hold]

Let $$ z = 2 + 2i$$ Find all complex numbers $w$ such that $w^4 = z$
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1answer
19 views

Factor complex equation

Im having some difficulty in factoring the following complex equation. The image bellow is taken from WolframAlpha, can anyone explain how I can factor this equation. In the task I am told one ...
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2answers
57 views

Can the cube of 2 different complex numbers be the same?

Can the cube of 2 different complex numbers be the same? I think it cannot be the same, but I don't really know how to prove it. I tried to expand it but it gives a very ugly result.
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2answers
38 views

How to draw the Bode diagram for a given transfer function?

With this transfer function: $$G(s)=\displaystyle\frac{10(s+1)}{s(0.1s+1)}$$ I need to do operations to draw the Bode diagram manually I have this: $G(jw)=\displaystyle\frac{10jw+10}{-0.1w^2+jw}$ ...
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9answers
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Is there an interval notation for complex numbers?

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$ ...
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2answers
69 views

Using complex solutions in a factorisation

I'm working through an assignment, and have become stuck understanding the question... In part (a) I am asked to solve the equation: $z^5 = -1$ I have done this, so I now have a set of solutions: ...
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1answer
49 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
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3answers
282 views

The limit of complex sequence

$$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$ I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow ...
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2answers
32 views

Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...
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0answers
28 views

Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..." If someone could clarify why that line is true I ...
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1answer
23 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
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4answers
81 views

Question on definitions

I was going through some basic recap of complex numbers and in the book (M. Boas. Mathematical Methods in the Physical Sciences) she says we define $e^{ix}$ by the Taylor series with $x$ replaced by ...
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1answer
46 views

When $f=u+iv$ is a holomorphic function, the real part of $f'(z)$ is equal to $u_x(z)$

Suppose $f$ is holomorphic, and is written as $f=u+iv$ with $u,v$ real-valued. Why is the partial derivative $u_x(z)$ equal to $\operatorname{Re}(f'(z))$? Source This fact is used in the proof ...
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2answers
63 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
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0answers
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Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
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1answer
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complex functions inequalities plane

Given $w(z)=\frac{i-z}{i+z}$. Find the map w=f(z) of the part of the plane defined by inequalities: $|z|>1$ and $Im(z)>Re(z)$ so far: $|z|>1$ is this area from $Im(z)>Re(z)$ => ...
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3answers
75 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
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3answers
45 views

The modulus of complex number? [closed]

the modulus of complex number $\frac{3 + 4i}{1 - 2i}$ is: $A) -\pi \qquad B) - \pi/2 \qquad C) \pi/2 \qquad D) \pi$ explain procedure also with your answer.
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1answer
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Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
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1answer
54 views

How to prove the formula for the argument of a complex number?

$\arg(x + iy) = 2 \cdot \arctan(\dfrac{y}{x + r})$ there is always the mark, this is derived from the 'Half-angle formula' How can I come from $\tan(\phi) = \tan(\phi + k \pi) = \dfrac{y}{x}$ to ...
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0answers
24 views

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups and orthogonal groups?

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups $Sp(2n, \mathbb{R})$ and orthogonal groups $O(n)$? My question may not be ...
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2answers
64 views

Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why.

I looked up the solution guide and found out: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49+4$ $=53$ Why the unknown "$i$" just disappeared$?$ I supposed it might be: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49-4i$ does it? ...
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2answers
37 views

Complex Numbers Closed under division?

My question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number.
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3answers
118 views

Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
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2answers
37 views

How to solve equation in complex numbers?

For $n$ odd ( e.g. with $n\equiv 1\mod 4 )$ I seek a solution $f(n)$ for this simple equation in the complex numbers $$(-1)^{f(n)}2^{\frac{n-1}{2}}=-\frac i2(1+i)^{n+1}$$ $f(n)$ is probably an integer ...
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0answers
22 views

On the criterion of convergence of infinite products of complex numbers

I have troubles in understanding the proof of the criterion which states that an infinite product exists iff the series of the complex logarithms of the terms of the product converges. In particular, ...