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

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Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
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complex numbers: determining whether claims are right or not.

we should decide whether the following claims are right or not, and explain our decision. let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$ a) $w_1^{1991} + w_2^{1991} + ...
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1answer
9 views

Solve ${z_1/\overline{z_2}} = z^3$

Let $z_1,z_2$ be complex numbers such that: $$z_1= 4\sqrt{2}-48\sqrt{2}$$ $$z_2= \cos{135^\circ} +i\sin{135^\circ}$$ Find all the complex numbers $z$ that fulfill the following equation: ...
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38 views

complex numbers

I have a number of questions about complex numbers and I need your help: z1, z2, z3, z4, z5 are complex numbers that fulfil |z1|=|z2|=|z3|=|z4|=|z5|=1 prove that |z1+z1+z3+z4+z5| = $|{1\over z1} ...
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complex numbers two problems

1) If $z=\cos\alpha+i\sin\alpha$ for $\alpha \in[0, 2\pi]$ then find $\alpha$ for $z^2+z$ I transform to this moment $\displaystyle ...
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Why is $t=\frac{1}{2}$ a root for $\tan 4\theta= \frac{4t-4t^3}{1-6t^2+t^4}=\frac{-24}{7}$, where $t=\tan \theta$

Show that $(2+i)^4=-7+24i$ $$\cos 4\theta = \cos^4 \theta - 6\cos^2 \theta \sin^2 \theta + \sin^4 \theta$$ $$\sin 4\theta= 4\sin \theta \cos^3 \theta- 4 \sin^3 \theta \cos \theta$$ ...
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1answer
14 views

z and w are two complex numbers prove the relationship

If $z$ and $w$ are complex numbers such that $|z+w| = |z-w|$ Prove that $\arg z - \arg w = \pm \ \pi/2$ Can someone please help me?
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1answer
8 views

finding equation of circle in complex plane

So i was asked to find the equation of the circle going through 1, i, and 0 Now i know that the equation of circle in complex form is: | z - (Zo) | = r where r is radius. So based on these values, ...
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1answer
45 views

How to find the roots of $(\frac{z-1}{z})^5=1$

Write down the fifth roots of unity in the form $\cos \theta + i \sin \theta$ where $ 0 \leq \theta \leq 2\pi$ Hence, or otherwise, find the fifth roots of i in a similar form By writing ...
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1answer
29 views

Describing a subset of the complex plane formed by z satisfying |z-i| + |z+i| = 3

I have been asked to describe the subset of the complex plane which is formed by the complex numbers z satisfying |z-i| + |z+i| = 3. It was easy to see that if the points z lie on the line segment ...
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2answers
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How to compute $(i^2-i^4+i^6-i^8+…+i^{38})^2$

How can i compute $(i^2-i^4+i^6-i^8+...+i^{38})^2$ ? I can see that the powers are arithmetic progression with $d=2$ but i tried to compute $S_{19}$ but it didn't work. Thanks.
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3answers
36 views

what is $c^{a+\mathrm i b}$ for $c \in \mathbb{R}$?

How can be $c^{a+\mathrm i b}$ for $c,a,b \in \mathbb{R}$ rewritten in the form $e+ \mathrm i d$ for $d,e \in \mathbb{R}$ (i.e. as a $\mathbb{R}$-linear combination of $1, \mathrm i$)?
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2answers
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maximum, complex quadratic function, Is my solutions correct?

I'm trying to compute $\max_{|z| \le 1} |(z+2)(z-1)|$. Here's how I do it: $\{z \in \mathbb{C} \ | \ |z| \le 1 \}$ is compact and $f(z) = (z+2)(z-1)$ is continuous, so it suffices to look for ...
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3answers
76 views

How to find the roots of $(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$

Write down, in any form, all the roots of the equation $z^5 − 1 = 0$ Hence find all the roots of the equation $$(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$$ and deduce that none of them is real ...
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Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
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Negative imaginary exponents

I was reading this question earlier: Understanding imaginary exponents In the answer, the answerer says $A^i=x+iy$ Furthermore, we can write $A^{−i}=x−iy$ for the same $x$ and $y$. Can ...
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1answer
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How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
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1answer
9 views

complex number conjugates (simple)

Show, by squaring both sides, that $|z - 10i| = 2|z-4i|$ is equal to $zz^* - 2iz^* + 2iz -12 = 0$ The bit I'm really stuck on (reading through the answers) is how $(z-10i)^2 $ is equal to $(z - ...
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1answer
19 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
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2answers
32 views

Real part of a complex number divided: $\Re\frac{z+1}{z-1}=0$

$\Re\frac{z+1}{z-1}=0$ I've tried so many methods, they all end up with two variables $a, b$. I tried setting $z= a+ib$. This give me the equality $2\cdot\Re\frac{z+1}{z-1} = ...
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Complex number arguments question

Given that u = -3i, how would I go about tackling these questions: (ii) For complex numbers 􏰀 satisfying arg(z􏰀 − u) = 0.25π, find the least possible value of |􏰀z|. (iii) For complex numbers 􏰀 ...
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1answer
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Finding the square roots of a complex number.

Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram. So far I've worked out the mod/arg form of the ...
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4answers
26 views

Solving a complex equation of the form $f(z)=0$

I'm trying to solve the following equation for $z\in \mathbb{C}$: $(z^3-64)(z^3+64)=0$. I'm not sure I'm doing it right, and need some guidance... I split this into two: 1. $z^3-64=0$ 2. $z^3+64=0$ ...
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3answers
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Solve complex equation $z^3 = i$

I have this $z^3 = i$ complex equation to solve. I begin with rewriting the complex equation to $a+bi$ format. 1 $z^3 = i = 0 + i$ 2 Calculate the distance $r = \sqrt{0^2 + 1^2} = 1$ 3 The angle ...
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1answer
20 views

Argand diagram intersection of point p

In an Argand diagram, the loci $Arg(z-2i)=\pi/6 $ and $ |z-3|=|z-3i|$ intersect at the point P. Express the complex number represented by P in the form re^iQ I try to sketch the Argand(sorry for ...
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1answer
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Show that $D_n$ is a subgroup of Perm($\mathbb{C}$).

For $ n \in \mathbb{N}$ and $0 \le r <n$, define $f_r : \mathbb{C} \to \mathbb{C}$ ; $z \mapsto ze^{2\pi i r/n}$ and $c: \mathbb{C} \to \mathbb{C}$ ; $z \mapsto \bar{z}$. a) Let $D_n = \{ f_0, ...
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Inverse cosine of a complex number, take $\cos z=\sqrt{2}$ for $z$

If I am given $\cos z=\sqrt{2}$ for $z$ and asked to solve it using the following: $$ \cos^{-1} z =-i \log\sqrt{z+i(1-z^2)} $$ I've only gotten as far as taking $\cos z=\sqrt{2}$ and changing it to ...
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1answer
20 views

Derivative of complex conjugate

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial ...
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1answer
37 views

Trying to evaluate $\prod_{k=1}^{n-1}(1-e^{2k\pi i/n})$ for my complex analysis homework

For my complex analysis homework, I am trying to show that the integral of the real function $1/(1+x^n)$, for integer $n\ge2$, along the positive real line is $$\int_0^{\infty}\frac{dx}{1+x^n} = ...
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Equations with modulus of a complex variable

I am struggling a bit to solve equations involving the modulus of complex variables. I am given the equation $|z-z_0|=|1-z_0z^*|$, where $z$ is a complex variable, $z_0$ is a complex number and $z^*$ ...
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Complex number $|z-w|$

On an Argand diagram, sketch the locus representing complex numbers satisfying $|z + i| = 1$ and the locus representing complex numbers w satisfying $\arg(w − 2) = 3\pi/4$. Find the least value of ...
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Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...
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1answer
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Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
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2answers
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How to solve higher grade polynomials of complex numbers $q^{10}-2q^5+2=0$

If I wanted to find the roots for $q^{10}-2q^5+2=0$, how would I go about doing that? I tried treating it like a quadratic equation, but couldn't get there. I also tried putting $q=(a+ib)$ but that ...
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Solving a Complex Number polynomial problem

This is an example Complex equations problem, everything is well understood except --(ii) in the below solution. Please can anyone explain, how anyone could have guessed the expansion in (ii) of the ...
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1answer
36 views

Solving $z^2-2iz+1=0$ in complex numbers

Solve: $z^2-2iz+1=0$ I did: $$(z-i)^2-(i)^2+1=0$$ $$(z-i)^2+2=0$$ $$((z-i)-\sqrt{2})((z-i)+\sqrt{2})$$ but that's wrong. Why?
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Study the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$

Study the character of the series $$\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$$ With i is the immaginary unit, $\theta$ is a real angle. My answer is that the series ...
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How to find magnitude of complex fractions function

Could you help me to find square of magnitude of complex fraction function that given by $$G=\frac {s+2}{s^2+2s+2}$$ where $s=j\omega$ Thank all This is my solution $$|G|^2=\frac ...
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2answers
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Factoring Polynomial with Complex Coefficients - Cauchy's Theorem

I'm faced with another polynomial (with complex coefficients) that I seem to only know how to solve using wolfram alpha. Here is the original integral that I need to compute using algebra and Cauchy's ...
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Algebraic expressions with complex coefficients $(1-i)z^2-2iz-4=0$

How does one solve expressions such as $(1-i)z^2-2iz-4=0$ Own attempt $$\begin{align} &z^2-\frac{2iz}{1-i}-\frac{4}{1-i}=z^2-z(1-i)-(2+2i)=0\\\iff&z^2-z(1-i)-\frac{i}{2} = 2+\frac{3i}{2} ...
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1answer
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Short question about Complex Numbers: $\forall z\in\mathbb{C},\exists\theta\in\mathbb{R}:e^{-i\theta}z=-|z|$

Is the following statement true? $\forall z\in\mathbb{C},\exists\theta\in\mathbb{R}:e^{-i\theta}z=-|z|$ I believe it is because if $z=|z|e^{i\alpha}$ then $\theta=\alpha-\pi$ should work?
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Draw a set of values

I have no idea how to draw properly those inequalities: a) $\left | \frac {z+3}{z-2i} \right | \geqslant 1$ b) $\left | z^{2}+4 \right |\leqslant \left |z-2i \right |$ While trying to solwe a) I got ...
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1answer
23 views

Calculate $\Bigl|\frac{1}{n^z}\Bigr|$.

If $z\in \mathbb C$ calculate: $$\Bigl|\frac{1}{n^z}\Bigr|$$ with n a natural number. And suggstion please? Thank you very much.
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41 views

Solve $e^z + e^i = 0$

Solve $e^z + e^i = 0$ Normally I would set $e^z = e^i$ and then use $z = i$ However in this case I have $e^z = -e^i$ which confuses me. Also I'm not sure if it's possible to go to $z = i$ since ...
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Finding an angle $\theta$ in a complex number

If we know that $z = \frac{1}{\sqrt2}(\cos\theta+i\cdot\sin\theta)$ and also that $z = \frac{(\sqrt3-1)+i(\sqrt3+1)}{4}$ How can I find $\cos\theta$ and $\sin\theta$? Using a calculator it gives me ...
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Question about convergence in complex numbers field

It may be a simple question, but if we want to show that $(z_n)\subset\mathbb{C}$ is convergent to $z\in\mathbb{C}$ then we should just check that absolute value of $z_n$ is convergent to absolute ...
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2answers
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Am I going about this wrong? Complex expression to polar form

I have the expression below, which I'd like to write in polar form. $$z = \frac{i}{{1+\frac{i(\sqrt3-1)}{1+i}}}$$ Own process My process was very tedious; and I also wouldn't solve the final part ...
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1answer
38 views

De Moivre's Theorem for proving

I have been asked by my lecturer to answer this question but I'm having problems doing so. The question is: Prove that $$\cos (5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5 ...
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3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
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2answers
27 views

How to determine roots of a complex polynomial [duplicate]

Let us consider an equation $x^3+10x^2-100x+1729=0$. Will this equation have at least one complex root having modulus $>12$?