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

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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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0answers
25 views

Show that $D_n$ is a subgroup of $\mathbb{C}$!

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 z conjugate$. a) Let $D_n = \{ f_0, ...
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12 views

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
16 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
25 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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18 views

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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0answers
14 views

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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3answers
187 views

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
25 views

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
22 views

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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2answers
20 views

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
33 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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0answers
37 views

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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0answers
10 views

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
23 views

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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2answers
26 views

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
22 views

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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2answers
14 views

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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3answers
40 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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3answers
18 views

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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2answers
16 views

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
15 views

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
36 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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17 views

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 $>12$?
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3answers
29 views

Calculate and describe the whole complex numbers group which…

Calculate and describe the set of complex numbers which: A) $$\frac{1}{Z} + \frac{1}{Z} = 1$$ B) $$|\frac{Z - 1}{Z + 1}| <= 1$$ Which steps should I follow? Any advices?
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1answer
16 views

Standard basis for Complex vector space

What will be the standard basis of $\mathbb{C}^3$ or in general how can I find the standard basis for $\mathbb{C}^n$ ? Note: $\mathbb{C}$ is complex vector space
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1answer
21 views

Simplification of $|a+b|^2$ for $a,b \in \mathbb{C}$

How do I simplify $|a+b|^2$, where $a,b \in \mathbb{C}$ and $|a|=|b|=1$? I know that the result is $4-|a-b|^2$, but I would like to be be explained how to do the simplification in the most elegant ...
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0answers
3 views

Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
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1answer
15 views

Draw a set of values in complex plane

I need help with that exuations: a) $\arg (z+2-i)=\pi$ b) $\pi \leqslant \arg [(-1+i)z]\leqslant \frac{3 \pi}{2}$ c) $ \arg \frac {i}{z}=\frac{3\pi}{4}$ I have more of them but I don't know how to ...
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1answer
25 views

Linearly Independent or Dependence of Complex Vectors — Homework Help [on hold]

Determine whether the indicated sets of complex vectors are linearly independent or dependent. $\left[\begin{array}{cc}i\\1\end{array}\right]$$\left[\begin{array}{cc}1\\i\end{array}\right]$ ...
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1answer
23 views

$\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$\{0}

I need to prove the set identity of the complex logarithm $\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$. ...
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1answer
31 views

Calculate complex number considering…

How can I calculate: $$ \frac{1-Z}{1+Z} $$ ...considering $Z = \cos(\alpha) + i \sin (\alpha)$ I have replaced the expression but I don't know how can I continue...
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3answers
35 views

Calculate value of a real number, considering “n” as a natural number

How could I calculate the value of the real number: $$ (1 +i \sqrt{3})^n + (1 - i \sqrt{3})^n $$ ...considering $n$ as a natural number and $i$ as the imaginary unit.
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1answer
14 views

the crossover point of four complex points

If there is four complex points $z_1,z_2,z_3,z_4$ in complex plane $\mathbb{C}$, I want to get the crossover point of the line $z_1z_2$ and $z_3z_4$. If I use the $Re(z_i)$ and $Im(z_i)$, it is easy ...
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2answers
63 views

What is meant by $(a+ib)^{c+id}$

I am currently studying complex numbers. Recently I saw terms like this: $(a+ib)^{c+id}$ , actually, I was simplifying them. But it was okay till I arrived at the following: ...
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1answer
20 views

Connectedness of a set of complex numbers

Is the graph of {$xy=1$} in $\mathbb C^2$ connected? I know $xy=1$ is disconnected in $\mathbb R^2$ by drawing its graph.But how to approach in $C^2$
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1answer
19 views

Complex number modulus identity (on unit circle)

For any three complex numbers $z_1, z_2, z_3$ on the unit circle, $|z_1 + z_2 + z_3| = |z_1 z_2 + z_1 z_3 + z_2 z_3|$. I am able to prove this by putting each number in modulus-argument form and ...
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1answer
23 views

Existence of a root $\alpha$ so that $|\alpha+i| <1$

For some monic polynomial $P(z) = \displaystyle \sum_{k=0}^n a_k z^k, 0 < |P(i)| < 1, a_k \in \mathbb{R}, k=0,1,...,n$, how does one show that a complex root $\alpha$ exists such that $|\alpha + ...
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2answers
25 views

Complex Cubic Equation z^3+3z+2i=0

How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?
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1answer
50 views

If $w = e^{2i\pi/5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$=?

If $w = e^{i\frac{2\pi}5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$ =? I substituted $w$ into the expression and combined similar terms. I then tried to ...
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2answers
32 views

Write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form $a+bi$

I have to write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form: $a+bi$. I think I should consider $z=\cos(k \pi / 6)+i\sin(k \pi / 6)$ and also use the fact that $\sum ...
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3answers
21 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
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1answer
15 views

A question on absolute values of line integrals

Let $f:U\rightarrow \mathbb{C}$ be continuous and $\gamma:[a,b]\rightarrow U$ be a smooth path where $U$ is open. Then we know that $$\int_{\gamma}^{}\ f(z)dz=\int_{\gamma}^{}\ ...
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5answers
53 views

How to find the complex solution of $x^6$

How do you find the complex solutions to $x^6+x^3-2=0 $ Obviously $x=1$ is one solution, but i cant get further than that.
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1answer
26 views

Prove that $\hat{f}(n)={\frac{2}{\pi(1-4n^2)}}$, given that $f(x)=|sin(\pi x)|$

Prove that $$\hat{f}(n)={\frac{2}{\pi(1-4n^2)}},\ given\ thatf(x)=|sin(\pi x)||,\int_{0}^{1}sin\pi(x)dx={\frac{2}\pi}\\where\ \hat{f}(x)=\int_{0}^{1}f(x)e(-nx)dx. \ Use\ the\ fact\ ...
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1answer
16 views

A question on branch of an inverse

Ignoring the 1st part of the 1st sentence of the question all I want to get is a branch $f$ of the inverse function of $g(z)=z^4. $ This is how I set about doing it, however, I need to verify this. ...
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2answers
32 views

Limit of $i^{n!}-2^{-n}$

I ran into this problem in Palka's Book which said to compute the limit of $z_n=i^{n!}+2^{-n}$. My approach was to consider the real and imaginary limits separately. Clearly the limit of the real part ...
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6answers
1k views

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$ Hint: solve for $b^2$ in terms of $a^2$ and then solve for $a$ I've attempted the question but I don't think I've done it correctly: $$ ...