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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91 views

What does it mean to multiply a real matrix by a complex scalar?

In this answer http://math.stackexchange.com/a/219508/27609 it is noted, that multiplying a matrix $A$ by a scalar $s$ is the same as multiplying a matrix $A$ by a diagonal matrix ${\rm ...
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3answers
54 views

Complex number and product of roots

For a second order ODE y''+10y'+ 21y=0 which was reduced to this quadratic expression x^2+10x+21=0 is there any way to tell whether the expression is bounded that is y(x) is either periodic or ...
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2answers
54 views

Limit concept under Complex analysis

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$
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1answer
25 views

Product of 2 couples of complex numbers

Let $a, b, c, d$ be complex numbers, but such that $b = \displaystyle \frac{a}{k}, d = \displaystyle \frac{c}{k}$ with $k$ real. Moreover, $$ab^* = \frac{|a|^2}{k} = cd^* = \frac{|c|^2}{k} = r$$ ...
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1answer
58 views

Complex No.s Proving Question

This is a problem from G.Stephenson's Mathematical Methods for Science Students which I am stuck on: Prove that, if $z = \cos \theta + i \sin \theta$ and $ n $ is any positive integer, $ z^n - \frac ...
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1answer
158 views

Does a purely imaginary number have a corresponding “angle” in polar coordinate system?

Let's say we have a pure imaginary number with no real part, $i$. I know that complex numbers in the form $a+bi$ can be converted into the polar coordinate system using the following relations: ...
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1answer
48 views

How to solve complex number

how to solve below complex number problem . The points $A,B,C$ represent the complex numbers $z_1,z_2,z_3$ respectively, and $G$ is the centroid of the triangle $ABC$ . If $4z_1+z_2+z_3=0$, ...
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1answer
29 views

Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
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1answer
32 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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1answer
117 views

Complex Number finding the general Value of Theta

If $$(\cos\Theta+i\sin\Theta)(\cos2\Theta+i\sin2\Theta)(\cos3\Theta+i\sin3\Theta) \dots (\cos n\Theta+i\sin n\Theta)=i $$ then show that general Value of $$\Theta=\left[2r+\frac1{n(n+1)}\right]\pi$$ ...
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1answer
43 views

Locus in Complex plane

Could someone help me out with this one Show that the locus of w as z varies with |z| = 1, where w is given by $$w^2=\frac {1-z}{1+z}$$ is a pair of straight lines.
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1answer
68 views

Complex numbers

If someone could help me with this question I would really appreciate it.For some reason I am getting a weaker version of these inequalities when applying triangle inequality. Let S be the interior ...
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2answers
78 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
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0answers
30 views

Rationalization of complex denominator of n-th power

I am stuck with rationalization of this expression, $$ \dfrac{(i\omega)^a}{x^a+(i\omega)^a}, $$ where '$\omega$' is frequency, '$x$' is constant, '$i$' imaginary unit, and '$a$' is non-integer ...
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1answer
106 views

Is the complex cosine function surjective?

Let $\cos z=\frac{e^{iz} - e^{-iz}}{2}$ be the complex cosine function. Then is $\cos:\mathbb{C}\rightarrow \mathbb{C}$ surjective? If so, how do i prove this?
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2answers
265 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
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2answers
90 views

Quadratic equation with “i” in discriminant

I'm solving some complex problems (pun intended), but I'm unable to solve any of this one type. One just has to solve a quadratic equation, but an imaginary number is in the discriminant. So to give ...
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0answers
66 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
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3answers
118 views

Every imaginary number is also a complex number?

How is it possible that every imaginary number (multiple of i ) is also a complex number?
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1answer
37 views

Explanation on how this solves for the real part of a complex fraction

I'm trying to solve the following fraction to find out what omega $\omega$ will leave me with only the real parts, assuming I know the values L, C, and R. $z = \dfrac{ \dfrac{L}{C} + \jmath\omega ...
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2answers
82 views

What is $\tilde{\Bbb{C}}$

$\tilde{\Bbb{C}}$ was defined in the following manner $\tilde{\Bbb{C}} = \Bbb{R} \cdot 1 + \Bbb{R} \cdot e$ with $1 \cdot 1 = 1, 1 \cdot e = e \cdot 1, e \cdot e = 1$ Could you elaborate more on ...
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2answers
55 views

Determining all complex Z in the equation

Let $n \in \mathbb N$. Determine all complex numbers $z \in \mathbb C $ such that $ |z| ^{n-2} = 1.$ How would i begin this question, thanks!
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2answers
59 views

Finding the Remainder of Complex Polynomials

Suppose $f(-1 + i) = 2 + 5i$ and $f(-2 - i) = -3$ determine the remainder of $f(x)$ divided by $(x + 1 - i)(x + 2 + i)$. I don't really know where to start any help would be great. Thanks :)
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1answer
14 views

Expressing two complex variables with constraints in terms of a single one without

I have two complex variables $u$ and $v$ with the constraints $|u|^2+|v|^2=1\ \ \ \ \ \ \ $ and $\ \ \ \ \ \ \ \ uv=|uv|\ $. With two constraints there are basically two degrees of freedom, and I ...
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0answers
61 views

For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
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1answer
57 views

What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.
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1answer
98 views

Show that for triangle ABC, with complex numbers for the coordinates, that we have the following equation

so I am doing an assignment on triangles and complex numbers, but I am stuck in the very first question. I am not asking for the solution, I would just like a hint or some ideas on what I need to look ...
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1answer
45 views

Complex Variable Proof Check

I have been stumped on this problem for a little bit, I figure that it's really simple and I'm missing something obvious, but I just wanted to see if what I have come up with is correct: Let z and w ...
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1answer
43 views

$z$ is a complex number, what is the solution of $z^n=-1$ for $n$ an interger and $\geq 2$

$z$ is a complex number, what is the solution of $z^n=-1$? For $n$ an interger and $\geq 2$. How can we expand $z^n$? Thanks in advance.
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1answer
49 views

Show inequality of complex number: $|\frac{a+b}{1+a\bar{b}}|<1$

Suppose $a,b\in\mathbb{C},|a|<1,|b|<1$, how to see $\displaystyle\left|\frac{a+b}{1+a\bar{b}}\right|<1$?
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1answer
45 views

What values or form of values can we get for these multiplications modulo a prime?

If we have four complex values, all of the form $a + b i$, for integers $a$ and $b$, we can label them $c$, $d$, $e$ and $f$. Now if we want to find $g$ and $h$ such that $$g \equiv ce \equiv df ...
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1answer
43 views

There is only one $z\in \Bbb C$ such that $z^2=w$ and $Re(z)>0$

Let $w\in\Bbb C$, show that unless $w\in \Bbb R^-$, there exists only one $z\in \Bbb C$ such that $z^2=w$ and $Re(z)>0$. This question is related to this other question, but this is a ...
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1answer
78 views

Extension to complex numbers

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.) EDIT: I'm mathematically ignorant, but I'm guessing such an extension can't be a ...
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1answer
134 views

Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
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1answer
75 views

Which basic operations are undefined even for complex numbers?

I'm aware of: $\frac{X}{0}$ (dividing by zero) $0^0$ (raising zero to the power of zero) Are there any others?
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1answer
65 views

$e^{i\theta} = \cos\theta + i \sin\theta$ for solid angle steradian , working mechanism.

$e^{i\theta} = \cos\theta + i \sin\theta$ for solid angle steradian , working mechanism. How will it work? For radian is 2D , i want for 3D.
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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
626 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
164 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
122 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
131 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
196 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
129 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
56 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
58 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
85 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
26 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 ...