# 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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### Product of distants from 1 to the corner of n gonal inscribe inside a unit circle.

Let $P_j=\cos \frac{2\pi}{n}+i\cos \frac{2\pi}{n}$ for $n\in \mathbb{N}.$ Show that $$\prod_{i=1}^{n-1}|P_j-P_0|=n.$$
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### Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
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### Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
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### Plot the numbers z in the complex plane that fufill $|z−2i|≤1$ and $\text{Im}(z){\ge}2$

I have recently started learning more about complex numbers and stumbled upon this problem: Plot the numbers $z$ in the complex plane that fulfil $|z-2i| ≤ 1$ and $\text{Im} (z) ≥ 2$ I know that ...
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### A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} + \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
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### Polynomial division for identifying an expression in terms of complex numbers.

This question is blatantly copied from here, for the sake of learning more I specify it a bit more: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = x+yi$$ I want to write $f(z)$ in terms of ...
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### Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has ..

Problem : Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has atleast a real root. No idea of approach : I am not getting idea how to approach a cubic polynomial in complex ...
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### Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Let $A\in M_{n \times n}(\mathbb{C})$. Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a characteristic polynomial for $g(A)$. I believe ...
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### Matrix and scalar multiplication

Say we have the following variables: A, a matrix that is nxn in size containing complex numbers B, a matrix that is also nxn in size containing complex numbers x, a scalar If you multiply, does it ...
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### Linear stability of an ODE $\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$

This is a part of exercise: Consider the following equation: $$\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$$ where all involved constants are ...
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### Neighbourhoods with proper multiplication

Assume we have two closed subsets $F$ and $G$ of $\mathbb{C}^*$ which are proper for the multiplication, i.e. $$KF^{-1}\cap G$$ is a compact of $\mathbb{C}^*$ when $K$ is a compact of $\mathbb{C}^*$. ...
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### number $-14+ \sqrt{15}+ \sqrt{12}$ which is located between two consecutive numbers? [on hold]

number of $-14+ \sqrt{15}+ \sqrt{12}$ which is located between two consecutive numbers? An interesting way , please
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### $2^z$ behavior when changing real and imaginary components of $z$

I'm reading The Music of the Primes by du Sautoy and I've come across a section that I'm having difficulty understanding: Euler fed imaginary numbers into the function $2^x$. To his surprise, out ...
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### Is this recurrence relation $g_{n+1}=ig_n-g_{n-1}$ is a trivial?

Let $g_1=i$ and $g_2=-1$, where $i=\sqrt{-1}$, and $$g_{n+1}=ig_n-g_{n-1}$$ For $n=1,2,3,4, ...$ then $g_n:={i, -1, -2i, 3, 5i, -8, -13i, 21, ...}$ respectively. Is this recurrence relation is ...
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### how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
### How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?
If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...