# Tagged Questions

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### Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
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### Under what conditions on $a,b$ is $1/(a+bi)=(1/a)+(i/b)$?

Question in proofs review in the complex numbers unit. I expressed $1/(a+bi) = (a-bi)/(a^2+b^2)$ I then separated the two terms in the denominator to get $a/(a^2+b^2)-bi/(a^2+b^2)$ I then equated ...
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### Find the number of distinct elements.

Let $\omega$ denote a non-real cube root of unity. Then find the number of distinct elements in the set $\{ (1+\omega + \omega^2 + \cdots + \omega^n)^m | m,n \in \Bbb Z_+ \}$
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### Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
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### Complex numbers exercise - homework

We have to prove that $z_1^{24n}+z_2^{24n}=2^{12n+1}$ if we know that a)$z_1z_2=2$ b)$z_1^3+z_2^3=-4$ I have tried many things but nothing worked so far
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### Explain why there are two complex numbers z such that $|z| = 1$ and that satisfy the equation $|z| = |z-1|.$

I must find both such complex solutions and express them in Euler form and usual form. So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. ...
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### Real and Imaginary [duplicate]

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that ...
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### Real and Imaginary

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that ...
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### Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
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### Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?
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### Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
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### Why $A\sin(2\pi ft) =\frac{A}{2j}(e^{j2\pi ft}-e^{-j2\pi ft})$ but not $\frac{A}{2}(e^{j2\pi ft}-e^{-j2\pi ft})$?

v_{\mathrm{in}}(t)=A\sin(2\pi ft) =\frac{A}{2j}\left(e^{j2\pi ft}-e^{-j2\pi ft}\right) \\ |H(f)|=|H(-f)|;\angle H(f) = -\angle H(f) \\ v_{\mathrm{out}}=H(f)v_{\mathrm{in}}=\frac{A}{2j}H(f)e^{j2\pi ...
let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
This is homework, so please don't give a full solution. Give a formula for $F_k$ for all $k$ where $f_n=4^n$ for all $n=0,\dots ,N-1$. I ended up abusing Wolfram Alpha and getting probably way ...