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

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0
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2answers
18 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
-1
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4answers
23 views

finding roots of cubic equation and the values of constants

$x^3+px^2+qx+30=0$ where $p$ and $q$ $\in R$, has a root $1+2i$. $1)$ Find the other non-real root. $2)$ Find the third root of the equation. Hence, or otherwise, find the values of $p$ and $q$.
2
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2answers
40 views

How can I solve the simultaneous equations that arise in solving $\cos(z)=2$.

If I have $\cos(z)=2$ I can say $\cos(a+ib)=2$ using double angle ideas $\cos(a)\cos(ib)+\sin(a)\sin(ib)=2$ using Euler's formula $\cos(a)\cosh(b)+i\sin(a)\sinh(b)=2$ equating real and imaginary ...
-3
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1answer
30 views

Solving $x^3 = -1$ for complex numbers [duplicate]

How can I solve for the complex solutions of $$ x^3 = -1 $$
0
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1answer
18 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
0
votes
1answer
24 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
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1answer
52 views

The value of $1+2\alpha+3\alpha^{2}+…+n\alpha^{n-1}$ for complex $\alpha$

Compute the value of $$1+2\alpha+3\alpha^{2}+...+n\alpha^{n-1}$$ in the form of a complex number where $\alpha$ is a non-real complex $n^{th}$ root of unity. The answer given is : ...
3
votes
0answers
25 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
0
votes
2answers
18 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
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2answers
39 views

What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument?

How is called the subset of Gaussian integers such that from all Gaussian integers having the same argument only one with the smallest absolute value is included? Is there a special name for them? ...
4
votes
2answers
42 views

Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.

Let , $f$ be entire function such that $|f\left(\frac{1}{n}\right)|\le \frac{1}{n^{3/2}}$ for all $n\in \mathbb N$. Then prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded. From the ...
0
votes
0answers
12 views

“Permutation” of squared norm and sum

In Problems and Solutions in Mathematics, 2nd edition, by Ta-Tsien, exercice 4312. Let $f$ be a periodic function on $\mathbb{R}$ with period $2 \pi$ such that $f|_{[0, 2 \pi]}$ belongs to $L^2(0, 2 ...
0
votes
1answer
32 views

Polynomial Equation Solution

Use Demoivre's theorem to show: $cos 7θ = 64 cos7 θ − 112 cos5 θ + 56 cos3 θ − 7 cos θ$ Hence,solve: $128x^7 −224x^5 +112x^3 −14x+1=0$ I've shown the first part and multiplied the equation by 2 and ...
0
votes
1answer
37 views

What is the number of complex integers inside a circle of radius r? [on hold]

What is the number of such complex integers, $z$, that $|z|\le r$? I am interested in a closed-form formula for integer $r$.
1
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3answers
49 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
1
vote
2answers
28 views

Expansion of imaginary numbers

If $(1+i)^{100}$ is expanded, what is the value of the real part of the result? I know that this has to do with binomial theory and Pascal's triangle, but I don't know how to use it here.
-2
votes
3answers
58 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
2
votes
0answers
24 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
0
votes
3answers
61 views

Simplifying a Complex Number

I have $\left ( \frac{e^{i\frac{\pi}{3}}}{1+i}\right )^{2014}$. I wish to simplify this to standard form. I simplify to $\left ( e^{i\frac{\pi}{12}} \right )^{2014}$ I can evaluate and simplify ...
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votes
2answers
70 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [on hold]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
0
votes
1answer
35 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
1
vote
3answers
46 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 ...
0
votes
2answers
37 views

Cauchy- riemann equations

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function that is differentiable at the point $z_0 =x_0 + iy_0$. Prove that $f'(z_0)= \frac{\partial u}{\partial x} (x_0,y_0) + i \frac{\partial ...
1
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0answers
37 views

How to use the for re^itheta to prove this?

Can someone please explain how to use the form $re^{i\theta}$ and de Moivre's to prove that: $$\sum_{n=1}^N \frac{\sin n\theta}{2^n} = \frac{2^{N+1} \sin \theta + \sin N\theta - 2\sin(N + ...
-1
votes
2answers
18 views

Representing a transformation from C to C with respect to the basis 1, i

I am having trouble understanding why the transformation: $ T(z) = (3+4i)z$ from C to C can be represented by the matrix $ \begin{bmatrix} 3, -4 \\ 4, 3 \end{bmatrix}$ with respect to the basis $ ...
0
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0answers
20 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
0
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0answers
24 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
1
vote
1answer
37 views

Prove that if $z$ is good then so is $z + r$ for every $r \in R$.

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ Note that $R$ is an integral ...
0
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0answers
14 views

An example of length, area or volume expressed as a complex number?

I sometimes have conversations with my fellow high school students about complex numbers and the existance of these "imaginairy" structures. I will then define the complex number $i$ algebraicly to be ...
2
votes
1answer
17 views

Use de Moivre's theorem to obtain an expression for $\sin^6x$ as a sum of terms in the form $\cos ax$

I'm not exactly sure if I'm on the right lines but I've started with a binomial expansion: $(\cos x+i\sin x)^6=\cos 6x +i \sin 6x= \cos^6 x + i(6\cos^5x \sin x)-15\cos^4x \sin^2x-i(20\cos^3x \sin^3 ...
2
votes
5answers
83 views

Argument of $z = 1 - e^{it}$

Let $t\in(0,2\pi)$. How can I find the argument of $z = 1 - e^{it}= 1 - \cos(t) - i\sin(t)$?
1
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1answer
26 views

Compute all possible values of log(-j)

How do I find all possible values of $\log(-j)$? I need to use the equation
5
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1answer
64 views

A small complex number whose total distance from other given complex numbers is large

Let $z_1,z_2,...,z_n$ be distinct complex numbers such that $|z_i|\leq1$. Is it true that there exists $z, |z|\leq1$ such that $\displaystyle\sum_{i=1}^n |z-z_i|\geq n$ ? Thank you.
0
votes
1answer
50 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
2
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0answers
32 views

Why is e used for polar form of complex numbers? [duplicate]

This is a real basic question. Why is $e$ the base for polar form of complex numbers? In high school maths I learned that e is useful in derivatives etc. And it's conventional to use it for ...
1
vote
4answers
41 views

Simplify $\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$

Simplify $$\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$$ I used de Morvre's theorem to get to $$\frac{(\cos ...
0
votes
1answer
30 views

What operation is “$\oplus$” in Lounesto's introduction to Clifford Algebras

I'm reading Lounesto's CLifford Algebras and Spinors and on page 26 (also below) he states the following: \begin{align} C\mathcal{l}_2=\mathbb{R}\oplus\mathbb{R}^2\oplus\bigwedge^2\mathbb{R}^2. ...
-3
votes
2answers
41 views

$Arg(z+1) = \frac{π}{6}$ and $Arg(z-1) = \frac{2π}{3}$ [closed]

I'm really stuck I need to find z when $$Arg(z+1) = \frac{π}{6}$$ and $$Arg(z-1) = \frac{2π}{3}$$ Please help!!!!
2
votes
6answers
69 views

If complex numbers can be represented as vectors, why can't we define $2$-dimensional vector division just as complex division?

If complex numbers can be represented as vectors, why can't we define $2$-dimensional vector division just as complex division? Is there any inconvenient/incompatibility to this?
2
votes
1answer
48 views

Jacobi Identities

Can anyone guide me how can I prove these two identities? a)$$\prod_{n=1}^{\infty}\frac{1-q^{2n}}{1-q^{2n-1}}=\sum^{\infty}_{n=1}q^{n(n+1)/2}$$ b) ...
1
vote
2answers
36 views

Prove equality of two numbers written in complex polar form.

Show that these two numbers are equal: $$ z_1=\frac{e^{\tfrac{2\pi i}{9}}-e^{\tfrac{5\pi i}{9}}}{1-e^{\tfrac{7\pi i}{9}}} $$ and $$z_2=\frac{e^{\tfrac{\pi i}{9}}-e^{\tfrac{3\pi ...
0
votes
1answer
39 views

Trigonometry question using complex numbers on the complex plane

I am not quite sure what this is asking, I tried to square these numbers and then convert into radians but it was not right. I am only used to graphing the absolute value of complex numbers. Let ...
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0answers
39 views

Challenge: Rotation by 1 radian [closed]

Prove in the most geometrical language you can (no Taylor series or pure algebraic manipulations) that $e^i$ represents a rotation by $1$ radian. Resorting to Euler's formula and identity are not ...
0
votes
1answer
28 views

Multiplication of two factors with complex numbers

I have the following to multiply ; $$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$ Now, I think that the product must not have any complex numbers... But here is what I get ...
1
vote
1answer
23 views

Evaluating complex functions integrals over closed curves

I recently evaluated the following two integrals: $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where $\gamma$ is a circle with radius $r$ around some point. $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where ...
0
votes
2answers
39 views

Argument for $(a+bi)^2$

I found out the modulus for $(a+bi)^2$, which is $$a^2+b^2$$ but I am unable to find the argument. I found out that $$\theta = \frac{2ab}{(a-b)(a+b)}$$ I don't know how to simplify further! Please ...
0
votes
3answers
22 views

Loci of Complex Equation

How does the loci of the equation $|z-(i+1)| = |1 + i|$ look like? I can't seem to visualise any points on the complex plane satisfying the above except the 2 obvious ones (2,2) and (0,0)... Is that ...
2
votes
1answer
28 views

If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then either $f\equiv0$ on $\Bbb C$ or $f(z)\not =0$ for all $z\in \mathbb C$.

Let , $f,g:\mathbb C\to \mathbb C$ be analytic such that $g(z)\not =0,\forall z\in \mathbb C$. If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then prove that either $f\equiv0$ on $\Bbb C$ or ...
2
votes
1answer
39 views

Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?

When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return is computed ...
0
votes
2answers
31 views

Where does this equality come from? complex numbers rewritten

http://mathfaculty.fullerton.edu/mathews/c2003/ComplexSequenceSeriesMod.html See example 4.2. in above. They have $z_n = (1+i)^n$ and then they've rewritten that to a familiar $a_n+ib_n$ form ... ...