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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235 views

Simplify formula $e^{-i0.5t}+e^{i0.5t}$

I want to ask, how can I simplify this formula ? $ e^{-i0.5t}+e^{i0.5t} $ I know that it can be simplify to $\cos(0.5t)$, but I don't know how :/
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2answers
35 views

Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $

The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation: $$ ...
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3answers
36 views

$|z_1+z_2|>|z_1-z_2|$ implies $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$

For two complex numbers $z_1$ and $z_2$, it is given that: $$|z_1+z_2|>|z_1-z_2|$$ How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$ If I take ...
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3answers
34 views

Complex conjugates

If $z=e^{2 \pi i/5}$, so that $z=x+iy$ where $x=\cos(2\pi i/5)$ and $y=\sin(2\pi i/5)$, then how are $z$ and $z^4$ complex conjugates with each other? I see that visually, but $$z^4=x^4-6 x^2 ...
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2answers
33 views

$z^4 = w$ has one solution $z = x + yi$ how do I find the other 3 solutions

As the title states, $z^4 = w$ has one solution $z = x + yi$ how do I find the other 3 solutions $x,y$ are real numbers
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2answers
36 views

How can I show that $e^{-in\pi}=(-1)^{n}, e^{-i\pi n/2}=(-i)^{n}$?

I practice some examples from a digital signal processing book. They used that $e^{-in\pi}=(-1)^{n}, e^{-i\pi n/2}=(-i)^{n}$. How did they come up with that? Of course I know Euler identity but I ...
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2answers
72 views

computing the value of $\sin^2(\frac{\pi}{10}$)

The question asks me to prove the following formula: $$\cos5\theta=\cos\theta(16\sin^4\theta-12\sin^2\theta+1)$$ which is pretty straightforward to do using De Moivre's theorem. They further ask me to ...
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2answers
22 views

Formula for $\arg_{\alpha}$

Define for $\alpha \in \mathbb R$ the complex function $\arg_{\alpha}$ which assigns to $z$ the unique value of $\arg z$ in $[\alpha, \alpha + 2\pi)$. Why is true that for any $z$, ...
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2answers
67 views

Find the Cartesian equation of the locus described by arg [(z-2)/(z+5)] = pi/4

My working: $$ \frac{x + iy - 2}{x + iy + 5} $$ $$ \frac{(x - 2 + iy)(x+5-iy)}{(x + 5 + iy)(x+5-iy)} $$ $$ \frac{x^2+5x-ixy-2x-10+2iy+ixy+5iy+y^2}{x^2+5x-ixy+5x+25-5iy+ixy+5iy+y^2} $$ $$ ...
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1answer
290 views

A proof that $|x+yi|=\sqrt{x^2+y^2}$, based on the given the conditions

If we attempt to define $|x+yi|$ by following conditions: $|x|=|xi|=x\operatorname{sgn}(x)$ (implicitly meaning the result will always be $\ge 0$) $|xz|=|x||z|$ $|z^x|=|z|^x$ for $x \in ...
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2answers
68 views

If $x=\omega-\omega^2-2$, then the value of $x^4+3x^3+2x^2-11x-6$ is?

If $x=\omega-\omega^2-2$, then the value of $x^4+3x^3+2x^2-11x-6$ is? ($\omega$ represents the cube roots of unity not equal to $1$). Directly substituting the given value will work. But there is ...
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2answers
47 views

About Euler's formula $e^{ix}=\cos x+i\sin x$ [closed]

I think I probably miss something. Can you tell me what it is? In my assumption, that any given 'x' value, $$e^{ix}=\cos x+i\sin x$$ But, why don't I get the same value in the equation when I ...
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2answers
64 views

Two complex numbers can be equal but why can't they are greater or lesser?

Yes we know that two complex numbers can be equal to one another , but why can't we say that a complex number is greater/lesser from another complex number ?
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5answers
58 views

Region on the complex plane: $|z-z_{1}| = |z-z_{2}|$. Intersection of two unit circles?

I have to draw the region on the complex plane defined by the following relation: $|z-z_{1}|=|z-z_{2}|$. After squaring both sides, we obtain the equality $(z-z_{1})\overline{(z-z_{1})} = ...
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3answers
65 views

When does $az+b \bar z +c =0$ have one solution?

When does $az+b \bar z +c =0$ have one solution? $a,b,c \in \mathbb C$ What I did: I rewrote: $a=a_1+ia_2$, $b=b_1+ib_2$, $c=c_1+ic_2$, and $z=z_1+iz_2$ Putting everything together we are ...
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3answers
173 views

how to find the cube roots of $-27i$? [closed]

i know to solve the question if it is given a+ib . but for this kind of question i can't solve it because it is only given 'ib' . for this type of question i am stuck on how to find the angle . could ...
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2answers
60 views

Find Complex Roots

Question: Find the complex roots of $$ {(z^{12} -1)\over (z^4-1)(z^3-1)} = 0 $$ What I have attempted: $$ {(z^{12} -1)\over (z^4-1)(z^3-1)} = 0 $$ $$ {(z^{6} -1)(z^{6} ...
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2answers
47 views

How to solve $(z+1)^3=-8i$?

I have a problem with this equation, I don't know what method to use. Can you show a method for the resolution? Thanks $$(z+1)^3=-8i$$
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1answer
32 views

Does an imaginary root,$\omega$ ,of the equation $x^n-1=0$ satisfy both $\omega=1$ and $\omega \ne 1$?

I am studying the $n$th roots of unity $1$ and I have a question about this exercise which is showed in my book: If $\omega$ is one of the imaginary roots of the equation $x^3=1$ ,then find the ...
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30 views

If $ { z_1 - 2z_2 }\over { 2 - z_1{\bar z_2} } $ is unimodulus and $z_2$ is not unimodulus then find $|z_1|$ .

$$ \left| {{z_1 - 2z_2 }\over { 2 - z_1{\bar z_2} } } \right| $$ $$ \implies | { z_1 - 2z_2 } | = | { 2 - z_1{\bar z_2} } | $$ I dont know how to proceed now .
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1answer
43 views

Raising i to a large power on a Ti-83

When I input this on my graphing calculator I get the correct answer: i^4 1 However when I do this with larger numbers I get a weird answer ...
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2answers
38 views

Express the following complex numbers in standard form

$$\left(\frac{\sqrt 3}{2}+\frac{i}{2}\right)^{25}$$ I know that you have to put it in the form $\cos\theta+i\sin \theta$ but I'm not sure how to go about it.
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2answers
174 views

The equation $z\sin z=50$

Let $\mathcal S$ be the set of all the solutions of the equation $$z\sin z=50\space\text{with}\space z\in \mathbb C$$ Try to make a description, as complete as possible, of the set $\mathcal S$, in ...
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4answers
71 views

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$ I'm not sure how to do this integration. It looks like partial fractions but I'm unsure.
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1answer
94 views

Why $0$ (zero) is a purely imaginary number? [duplicate]

I was reading this Wikipedia article and found that $0$ is a purely imaginary number. Why? Is it because $i0=0$? So zero is the only number which is real as well as purely imaginary? Any explanations ...
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3answers
27 views

Absolute value of a complex number with a arbitrary basis

I want to calculate the square of the absolute value of a complex number $x^{ia}$, with $x$ and $a$ being real while $i$ is the imaginary number: $$\left|x^{ia}\right|^2=?.$$ I have trouble because ...
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2answers
75 views

$Ax = kx$ for complex scalar $k$ and complex vector $x$, and symmetric matrix $A$

Prove that if $Ax = kx$ for some non-zero, possibly complex valued scalar $k$ and non-zero complex valued vector $x$, and real symmetric $n$x$n$ matrix $A$, then $k$ must be real valued. A hint was ...
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3answers
34 views

Express $u(x,y)+v(x,y)i$ in the form of $f(z)$

I need to express $f(z)$ from the form $\color{blue}{u+vi}$ to the form $\color{blue}z$ for example if: $g(z)=\frac{1}{x+yi}$ so $ =g(z)=\frac{1}{z}$ $$f(z)=\underbrace ...
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3answers
95 views

Find all the real solutions to the equation: $(x+i)^n-(x-i)^n=0$

Find all the real solutions to the equation: $$(x+i)^n-(x-i)^n=0$$ The answer is given, I will type it out until the line which is unclear to me (meaning I understand all the steps leading up to the ...
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3answers
60 views

Evaluating $\int_{0}^{2\pi} \exp({inx})\mathrm dx$

Can someone tell me why this integral gives zero: $$\int_{0}^{2\pi} \exp({inx})\mathrm dx$$ Where $n=1,2,3, \cdots $. I am a bit confused since $\int_{0}^{2\pi} \exp({nx}) dx$ does not give zero.
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1answer
66 views

Solve complex equation $5|z|^3+2+3 (\bar z) ^6=0$

I'm stuck in trying to solve this complex equation $$ 5|z|^3+2+3 (\bar z)^6=0$$ where $\bar z$ is the complex conjugate. Here's my reasoning: using $z= \rho e^{i \theta}$ I would write $$ ...
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3answers
50 views

Proving $|z_1z_2|=|z_1||z_2|$ using exponential form of a Complex Number

Problem: Prove $$|z_1z_2|=|z_1||z_2|$$ where $z_1,z_2$ are Complex Numbers. I tried to solve this using the exponential form of a Complex Number. Assuming $z_1=r_1e^{i\theta_1}$ and ...
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1answer
56 views

Find all the complex numbers that satisfy this quotient.

A certain problem that I have been working on involves the equation $$1 = \frac{1}{1-n}$$ One can see that the only real-number solution is $n=0$. As far as the original problem goes, that is ...
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2answers
133 views

Complex numbers - roots of unity

Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 ...
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4answers
194 views

Find all complex numbers $z=a+bi$ such that $z^3=8$.

Find all complex numbers $z=a+bi$ such that $z^3=8$. I'll be happy if someone say me with what steps I have to start solving this problem.
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120 views

Where's the mistake in this calculation? [duplicate]

Obviously something is wrong with this, but where is the error and why is it one? $$ \begin{align} \sqrt{-1} &= (-1)^{1/2} \\ &= (-1)^{2/4} \\ &= \sqrt[4]{(-1)^2} \\ &= \sqrt[4]{1} \\ ...
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38 views

Product of roots of unity using e^xi

Find the product of the $n\ n^{th}$ roots of 1 in terms of n. The answer is $(-1)^{n+1}$ but why? Prove using e^xi notation please!
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68 views

Shortcut Technique for finding Raised Binomials with Imaginary Numbers

Find the Value of $(1+i)^5$ where $i$ is an imaginary number. The answer is $-4\cdot (1+i)$ We can always multiply them manually; but $i$ was wondering if there are any math tricks to quickly ...
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2answers
51 views

Visual representation of complex polynomial

I am doing an exercise on stating whether a set is open, closed, its interior, boundary, etc in the complex plane and i have been relying on my geometric intuition and drawing diagrams in order to get ...
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2answers
177 views

Finding the orthogonal complement of a complex subspace

Let $i := \sqrt{-1}$ . Consider $W \subseteq \mathbb{C}^3$ defined by $W := \{(1, 0, i),(1, 2, 1)\} $. Find $W^\perp$. My biggest issue with this problem is not knowing how to extend the basis of $W$ ...
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42 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
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1answer
129 views

Proof of Lagrange Identity

I need to prove Lagrange Identity for complex case, i.e. $$ \left( \sum_{i=1}^n|a_i|^2 \right)\left( \sum_{i=1}^n |b_i|^2 \right)-\left| \sum_{i=1}^na_ib_i \right|^2=\sum_{1\leq i<j\leq ...
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3answers
54 views

Is a matrix with complex entries invertable?

This is merely a question of interest and not for something I am doing in school. I have never seen a matrix with complex entries in class before, but mind you it was a limited linear algebra class, I ...
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2answers
43 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
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2answers
162 views

Is $-e^{i\pi} = 1$?

Since $e^{i\pi} = \cos \pi + i\sin \pi = -1,$ a suspicious argument is to proceed to conclude that $$-e^{i\pi} = 1.$$ However, this leads to $$-e^{i\pi} = e^{0}.$$ Is the above reasoning wrong?
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3answers
218 views

Show f(z)/z is bounded

if $\lim\limits_{|z| \to \infty} \frac{f(z)}{z} = 0$ then how do I show that f is bounded. Intuitively, this makes sense to me but I having trouble writing it out formally. I was thing for $|z|>N$, ...
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1answer
50 views

$\sum\limits_{n=1}^\infty\dfrac{cos({2nt})}{n2^n} =$? for any real $t$

given that $\sum\limits_{n=1}^\infty\dfrac{z^{2n}}{n2^n} = -\log\left(1 - \frac{z^2}{2}\right)$ , could you calculate the sum in the title for every real $t$ ?
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3answers
81 views

Proving cot identity using Euler identities

I'm trying to prove that $\cot(2\theta)+\csc(2\theta)=\cot(\theta)$. I'm using that $$\sin(\theta)=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})\qquad \cos(\theta)=\frac{1}{2}(e^{i\theta}+e^{-i\theta})$$ So ...
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3answers
412 views

Find all complex numbers satisfying the equation

Find all complex numbers $z$ satisfying the equation $$\left|z+\frac{1}{z}\right|=2.$$
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1answer
350 views

Assume |f(z)| is constant for all z in $\Omega$ then f is a constant function

Here's what I got so far: Let $f(z)=f(x,y)=x+iy$. If $|f(z)|$ is constant then $\sqrt{x^2+y^2}=C$, a constant. I know that in the very end I need to show that $y=0$ for $f$ to be a constant $x$. ...