# 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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### 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 $z$...
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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? [closed]

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 ...
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### 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 ...
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### Stereographic projections.

1) Stereographicly project $\arg z =\frac\pi4$ 2)Using Inverse Stereographic projection map the 30th parallel south. My solutions: 1) That's a semi-cricle defined by $X^2+Y^2+Z^2=1;\ X=Y;\ X,Y>0$...
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### If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7})$ then construct a quadratic equation.

If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7})$, then find a quadratic equation whose roots are $\alpha = a + a^2 + a^4$ and $\beta = a^3 + a^5 + a^7$ . Using the fact that sum of $7th$ ...
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### How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
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### Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove $A^n=0_n$. ...
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### How to understand $|1+z|<1$ where $z\in\mathbb{C}$ geometrically?

I am trying to understand how to plot $|1+z|<1$ where $z\in\mathbb{C}$, is it a circle centred at real axis $\text{Re}(z)=-1$ with radius $1$? My question is how can we show it algebraically? I ...
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### isolating x with two variables and negative exponents

I have: $$4^y = x^{-2}$$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
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### Higher degree polynomial with complex roots

I'm working on the following problem: $$r^4 - 3r^2 -4r = 0$$ I factor out one $r$ and leaving me $r(r^3 - 3r -4) = 0$. One real root is $r=0$, and I'm unable to find the other ones. I tried ...
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### calculating $\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$ using Residual Theorem [duplicate]

Could anyone help me provide a way to calculate $$\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$$ using the Residue theorem in complex analysis? Many thanks
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### Complex projective line homeomorphic to $2$-sphere

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...
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### Express $\sin^3x$ in terms of cosines of multiples of $x$

I am studying complex numbers, and I have been trying to figure that out. Just not getting it. I keep getting $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
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### Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? f\left(a z\...
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### Unexpected result from Euler's formula

I am a bit confused with a result I get from Euler's formula: $e^{2\pi i} = 1$ $\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$ $(e^{2\pi i})^{\frac{1}{3}}= 1$ $e^{\frac{2}{3} \pi i} = 1$ This last ...
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### Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
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### The singular points and residues of $\sin(\frac 1 z)$

I met a question asking all the singular points and corresponding residues of $$\sin \frac 1 z$$ My understanding is that $$\sin \frac 1 z=\frac 1 z-\frac 1{3!z^3}+\frac 1 {5!z^5}+...$$ Thus ...
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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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### 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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### Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
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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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### 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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### Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} &...
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### complex numbers and locus

When the problem says that the complex number $z$ moves on the straight line $y=2x$,what "clue" do I get from that? And generally when it says that a complex number belongs/moves to a conic section ...
### 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 ...
### 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 $a+bi$.