# Tagged Questions

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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### An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$|z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2$$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!
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### Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
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### Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
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### Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}}$ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
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### Complex Matrix Representation

Lets say if $X\in C ^{m\times n}$, it does have real and imaginary parts. If I want to represent a matrix in real and imaginary form then why I write it this way where is $i$? \begin{bmatrix} X_r ...
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### Proving analytic function $f = 0$ under certain assumtions

I was given the following exercise: Let $f(z)$ be analytic in an open and connected set $U$ containing the point $z=0$ and assume $|f(1/n)| < \frac{1}{2^n}$ for $n \in \mathbb{N}_{> 0}$. Prove ...
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### How to find the absolute value of this complex number: $\frac{-4-6i}{17+i}$

I know that, in general, $|a+bi|=\sqrt{a^2+b^2}$, however, I don't know how to make $\frac{-4-6i}{17+i}$ into the form of $a+bi$.
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### Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
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### Solving systems of linear equations with complex numbers by hand

How can I solve a 3x3 system of linear equations with complex numbers by hand without making a mistake? I know that I can solve them either with Gaussian Elimination or Cramer's rule, but I find it ...
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### Square roots of complex number in exponential form

The complex number $z$ is defined by $z=\frac{9\sqrt{3}+9i}{\sqrt{3}-i}$. Find the square roots of $z$, giving your answers in the form $re^{i\theta}$.where $r>0$ and $-\pi < \theta \leq \pi$. ...
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### For complex number $z_1$ and $z_2$ determine angle $z_1 O z_2$

Let $w=\frac{\sqrt{3}+i}{2}$ and $P=\{w^n:n=1,2,3,.... \}$ Further $H_1=\{z \in C: Re(z)>\frac{1}{2} \}$ and $H_2=\{z \in C: Re(z)<-\frac{1}{2} \}$, where $C$ is the set of all complex numbers. ...
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### Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate?

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} ...
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### Solving for complex power

I am asked to find $(1+i)^{3+4i}$ This is what I have and I wanted to know if it is correct: $e^{3+4i [ln\sqrt{2} + i (\pi/ 4 +2\pi k]}$ $e^{3+4i ln\sqrt{2} + 3-4 (\pi/ 4 +2\pi k)}$ by ...
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### Closed contour integral: $\int_{\mathbb{c}}\frac{ z}{2z^{2}+1} dz$ where the contour is the unit circle

first and foremost please excuse my English. given $∫_c \frac{{z}}{2z^{2}+1}dz$ where the contour is the unit circle. so c = $e^{it}$ from 0 to $2\pi$. since the contour is the unit circle we can ...
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### What is the trigonometric form of the complex variable $z=0+0i$?

I'm confused how do i determine the trigonometric form of the complex variable $z=0+0i$ , it has modula such that is 0 but what about it's argument ? Note : At a least i would like to know it's ...
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### Why are There No “Triernions”? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
True or false: the polar coordinates of $-1-i$ are $-\sqrt{2}\operatorname{cis}\frac{\pi}{4}$ In my opinion it's true: $\tan\theta=\frac{-1}{-1}=1\Rightarrow \theta=\frac{5\pi}{4}$, ...
Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...