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

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1answer
34 views

Complex numer equation

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ . I'm not sure if I'm doing this question right, but would the solutions be $+ 1,-1$ or $0$?
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2answers
98 views

Complex number proof

Let f(x), g(x) $\in \mathbb C[x].$ Prove that if f(x) | g(x) and g(x) | f(x), then there exists a nonzero $c \in \mathbb C$ such that $f(x) = c * g(x)$ (You may use the fact that for any p(x), q(x) ...
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2answers
73 views

Complex Number Question - $|z^{z}|$

Find all possible values of $$\mid z^{z} \mid$$ using the polar for of $z$. I have tried putting it into polar form but nothing comes out that seems easy to work with/looks like a reasonable simple ...
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4answers
531 views

Find all roots of $x^{6} + 1$

I'm studying for my linear algebra exam and I came across this exercise that I can't solve. Find all roots of polynomial $x^{6} + 1$. Hint: use De Moivre's formula. I guessed that two roots are $i$ ...
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2answers
126 views

finding $\tan^{-1}(2i)$

I'm having some trouble finding $\tan^{-1}(2i)$. The formula the book has is $\tan^{-1}z=\dfrac{i}{2} \log\dfrac{i+z}{i-z}$. But when I use this I get a different answer than what the book has. This ...
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3answers
107 views

cubing the expression of a complex number

Calculate the solutions to $$\left(-8-8\sqrt{3}i\right)^3$$ I would really appreciate if you could help me with this. Thanks
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1answer
35 views

Complex Transformation

$z_1 = 1 + i$ and $z_2 = -1 + i$ I am told: $w = \dfrac{az + b}{z + d}$ where $z \not= -d$ Where a, b and d are complex numbers, maps the complex number $z$ onto the complex number $w$. Given that ...
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1answer
130 views

Remainders with complex numbers

Let $ f(x) \in C [x] .$ Suppose $ f(-1+i) = 2+5i $ and $ f(-2-i)=-3. $ Determine the remainder of f(x) divided by $(x+1-i)(x+2+i). $ How would i begin with this question, like how would i ...
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2answers
194 views

Determine all complex numbers z in equation:

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ How would I begin this? Would I begin by saying $z=a+ib$ and expand and stuff?
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1answer
121 views

Using Liouvilles theorem to show that f is identically constant on all of $\mathbb C$

Use Liouvilles theorem and the fundamental theorem of calculus to prove that for an entire function $f$, if there exists $M \in \mathbb R: Re(f(z)) \leq M$ $ \forall z \in \mathbb C $, then $f$ is ...
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2answers
55 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
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2answers
57 views

complex expression to the power of a complex expression

I have a math exam tomorrow, and i am not sure with my solution for a exercise. can you please tell me if i am right. Question is: $$(1+i)^{(1-i)}$$ My solution is: $$\sqrt{2} e^{(i ...
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1answer
74 views

Determining the number of complex roots (including multiplicities) of a polynomial

Could someone please explain/show me how to determine the number of complex roots including multiplicities of a polynomial such as $P(z):= 5i z^{37} - (6 +2i)z^{4} + 4z^2 - i$ Would i need to ...
2
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1answer
25 views

approximate a vector of complex numbers

Given a vector of complex number $\vec{z}=(z_1,\cdots, z_n)$ with $|z_i|=1$ and $z_i$ is not a root of unit, and a vector of complex numbers $\vec{r}=(r_1, \cdots, r_n)$ with $|r_i|=1$. Is it the case ...
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8answers
235 views

Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

I believe my trouble is that the identity, $e^{i \pi} = -1$, comes down to the definition of the exponentiation of $i$, which seems rather obscure to me. This is my current understanding of ...
2
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3answers
129 views

Sketching on the complex plane $|z + 1| - |z - 1| = 4$

Sketch the set of points that satisfy $|z + 1| - |z - 1| = 4$ on the complex plane. Wolfram alpha gives me an empty graph, I end up with the equation of an ellipse but with the condition ...
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2answers
86 views

Factoring $x^4 - x^2 + 1$

I'm interesting in finding the possible quadratic factorization of this polynomial: $x^4 - x^2 + 1$. My first idea was to do long division by $x^2+1$, but I did get a remainder, so I presume this ...
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4answers
58 views

roots of complex numbers $z^2$

Let $z^2 = \frac{1}{2} + \frac{\sqrt{3}}{2i}$ Where $z$ is an element of the Complex Numbers Find the two possible values of $z$ I would really appreciate if someone could help me with this ...
4
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6answers
215 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
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2answers
95 views

Factorising a complex polynomial over $\mathbb{C}$

I'm given $f(z)=z^6-1$ to factorise over $\mathbb{C}$. My working is as follows up to the point I don't understand: $f(-1)=0$ and $f(1)=0$ So $(z+1)$ and $(z-1)$ are factors ...
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2answers
189 views

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 ...
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1answer
58 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
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3answers
295 views

Solving a complex number equation with both $z$ and its conjugate $\bar z$

Determine all possible values of $z\in\mathbb{C}$ that satisfy the equation $4z = \overline{z}^2$. Where $\overline{z}$ represents the complex conjugate. (Hint: There are $4$ solutions.) ...
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2answers
107 views

Shortest distance formula in complex numbers

Let $L$ be a line in $\mathbb{C}$ that makes an angle $\alpha$ with the real axis. Let $z=x+iy$ be any point on the line $L$. Let $d$ be the shortest distance from $L$ to origin. Prove geometrically ...
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1answer
90 views

General formula for a square in complex numbers

I need to find a general formulae for a square, with its interior included, in terms of complex numbers. Note that your general square should have (general centre, side-length and orientation.) I do ...
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1answer
58 views

Does the complex conjugate just switch the sign in-between the numbers?

I have a question about imaginary numbers and their complex conjugate. My teacher denotes the complex conjugate to have a bar over it. For example: $\overline{3+5i} = 3 - 5i$ But does the complex ...
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2answers
57 views

Infinite field extension but algebraic subset

Is there a subset $ S \subseteq \mathbb{C}$ so that all $s \in S$ are algebraic over $\mathbb{Q}$ and $\left[\mathbb{Q}[S] \colon \mathbb{Q} \right]=\infty$? $\mathbb{Q}[S]$ is defined as ...
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1answer
75 views

How do we explain the existence of complex conjugation?

We can define the complex numbers by writing $\mathbb{C} = \mathbb{R}[i]/(i^2+1),$ where $\mathbb{R}$ is to be regarded as a commutative ring. Furthermore, since $\mathbb{R}$ happens to be a field, ...
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2answers
166 views

convert complex number in denominator of fraction into polar form

I got this in classroom $$Re\left[\frac{1}{j2\pi f_0 RC + 1}Ae^{j2\pi f_0t}\right]$$ $$=Re\left[\frac{1}{\sqrt{4\pi^2 f_0^2 R^2C^2+1}}e^{-j{tan^-1}2\pi f_0 RC}Ae^{j2\pi f_0t}\right]$$ I attend ...
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1answer
40 views

A complex log question

I'm trying to find the solutions of $\log(z)=i\log(\bar{z})$ where $\bar{z}$ is the conjugate of $z$. I'm aware of the multivalued complex log, so $\log(z)=\log|z|+i\arg(z)$ but I don't see to be ...
2
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2answers
47 views

Problem converting to polar form in proof

I wonder if anyone has an idea about how to write $$ \prod_{\substack{j=0\\ j\neq k}}^{n-1} ( e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi ij}{n}})=n,\qquad k=0,1,...,n-1,\; j=0,1,....n-1$$ in a "general" ...
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3answers
145 views

How to find number of real and complex roots?

Below is a question asked in JNU Entrance exam for M.Tech/PhD. I want to know if there is a fixed way to calculate it. I have failed to use the factor theorem. ...
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2answers
69 views

The real and imaginary parts of this complex number

The number is $(i+1)^{(i-1)}$ I tried but couldn't solve it.
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2answers
99 views

Paradox with function representation

Let assume the function $\eta(E)$ has the following representation: $$\eta(E) = \sqrt{\frac{a}{E}}$$ where $a$ is the known positive constant, and $E \in [-\infty, +\infty]$. I know that $\sqrt{a} = ...
4
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4answers
565 views

Compute the square root of a complex number

This is a follow up to a previous question. I solved the equation $z^4 - 6z^2 + 25 = 0$ and I found four answer to be $z = \pm\sqrt{3 \pm 4i}$. However someone in the comment said that the answer is ...
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4answers
62 views

showing that $(z_1^2z_2)^2$ is a real number

Given $z_1=a+bi,z_2=c+di,\frac{b}{a}=\frac{d}{c}=\frac{1}{\sqrt3}$, $a,b,c,d$ are real numbers; $z_1,z_2$ are complex numbers. Need to prove that $(z_1^2z_2)^2$ is a real number. So i figured that ...
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4answers
3k views

Absolute value of complex exponential

Can somebody explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook says.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
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1answer
92 views

How to solve quadratic function with degree higher than two?

I am struggling to solve the function $z^4 - 6z^2 + 25 = 0$ mostly because it has a degree of $4$. This is my solution so far: Let $y = z^2 \Longrightarrow y^2 - 6y + 25 = 0$. Now when we solve for ...
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1answer
62 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
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2answers
60 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
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4answers
105 views

Prove there is no complex z such that $|z|=|z + i\sqrt5| = 1$

This is a question in introduction to pure mathematics. I am pretty sure I am close to the answer but I can't quite decide why this proves that there is no complex numbers: $$|z| = |z + i√5| = 1$$ ...
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1answer
102 views

Example of holomorphic function from unit disc to itself

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 $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
2
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1answer
127 views

Cauchy's Theorem and Cauchy's formula

I came across the following problem in our last midterm exam. I am completely stuck as to how to begin the solution: If $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then $|f|$ has no ...
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2answers
60 views

Product of $1-\operatorname{cis}(2k\pi/n)$

I'm in a question about polygonals and got stuck at a part. I have to prove that $$\prod_{k=1}^{n-1} \left(1 - \operatorname{cis}(\frac{2k\pi}{n})\right) = n$$ I've tried to multiply it to make ...
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2answers
63 views

What happens if to postulate that complex numbers whose argument differs by $2 \pi$ are not equal?

What happens if to postulate that complex numbers whose argument differs by $2\pi$ are not equal? What properties such system will have? Will all analytic functions be entire?
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1answer
114 views

How to find all pairs $(a, b)$ s.t. $(a^2+b^2)/\gcd(a,b) \leq n$ for constant $n$?

Any help is appreciated, this is for my work on http://projecteuler.net/problem=153. Also posted here
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2answers
88 views

Using the polar form of $1 + i$ and $\sqrt3 + i$ to deduce $\cos (\frac{\pi}{12}), \sin(\frac{\pi}{12})$

I have been beating my head against the following problem and would like a gentle nudge in the right direction. The question states, by writing $1 + i$ and $\sqrt3 + i$ in polar form, deduce that ...
0
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2answers
53 views

Complex and Trigonometric Identities

How can I get this result: $$\frac{1+cis\theta}{1-cis\theta}=-\frac{1}{i\tan(\theta/2)}$$ I've tried to expand $1-cis\theta$ as $(1+cis(\theta/2))(1-cis(\theta/2))$, but it doesn't help.
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4answers
50 views

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$?

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$? Meaning, if I have a complex number and I multiply it by its complex conjugate, would that always return a number in $\mathbb{R}$? ...
2
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1answer
76 views

A problem on Möbius function

Here is an exercise from the text book "Combinatorics" (J.H.van Lint, R.M. Wilson) Let $f_n(z)$ be the function that has all its zeros as all the numbers $\alpha$ for which $\alpha^n=1$ but ...