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

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Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic number $\beta$ Pisot-like if $|\beta| > 1$ and all its conjugates lie inside the complex unit circle (here $|\cdot|$ is the usual absolute ...
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2answers
48 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 ...
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1answer
19 views

For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

I used the root test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the ...
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2answers
20 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 ...
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4answers
24 views

finding roots of cubic equation and the values of constants [on hold]

$x^3+px^2+qx+30=0$ 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 $q$.
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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} ...
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2answers
36 views

Which $k$ value from $2\pi k$ do you pick as the $n$th root of the solution?

$$W = \frac{1+i}{\sqrt{2}}$$ I need to find 5th root of $W$ where $Z^5 = W$ $\theta$ is: $$\frac{\pi}{20} + \frac{2\pi k}{5}$$ I always thought You need to plug in $K = 0, \pm 1, \pm 2,\ldots$ to ...
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2answers
41 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 ...
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1answer
230 views

Complex conjugate of a variable raised to the power $n$

What would be the complex conjugate for these three. Assuming $i$ is always $${\sqrt{-1}}$$ $$i^{11}$$ $$(2-3i)^3$$ $$\frac{3-i}{2i+5}$$
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1answer
83 views

Compute all possible value of a complex number

Let C be the set of complex numbers and j the imaginary unit. Compute all possible values of ((1 + $((1 + \sqrt{-3})^3 \times (1-j)^2)^\frac13 + j$ So, i tried doing the 2 ways, 1) = (1+$\sqrt3j) ...
3
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1answer
34 views

Convergence of a complex function

I need to proof if the following function is bounded and convergent. $f(n)=\left(\frac{10+in}{n^{2}+2in}\right)^{n}$ Status: This should be correct. Can anybody confirm this? I tried it with ...
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1answer
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1answer
77 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
5
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1answer
72 views

Solve complex equation with exponential

I have to solve: $$e^{3z}+3ie^{2z}-ie^z+3=0$$ My attempt: Let $0\ne x:=e^z$. Then we can rewrite our equation as: $$x^3+3ix^2-ix+3=0$$ $$ix^2(-ix+3)+(-ix+3)=0$$ $$(-ix+3)(ix^2+1)=0$$ So $x\in ...
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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 : ...
0
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1answer
80 views

complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $ 0\le \arg(z+2)\le 45\deg $ I was able to sketch and shade the region that satisfies both ...
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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 ...
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0answers
28 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 ...
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3answers
51 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 = ...
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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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7answers
376 views

Expansion of complex equation.

Find the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than ...
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2answers
49 views

Solve the complex equations

I have a question from complex calculus. How to solve this two equations: a) $$ \sin(z)=2015 $$ I know that $\sin(z)$ equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
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3answers
62 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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1answer
24 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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2answers
40 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? ...
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1answer
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What are other examples of complex associative operators besides, x + y +rxy, rxy, and x + y + 1/r?

I have been having fun (and frustration) in finding complex associative operators over the complex numbers. So far, I have found the 3 listed in the title (r is a constant), and also know about ...
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1answer
33 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 ...
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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 $r$.
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3answers
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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, ...
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2answers
30 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.
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2answers
54 views

Question about open mapping theorem

Let, $f:\Omega\to \mathbb C$ be a non constant anlytic function on an open set $\Omega \subset \mathbb C$.For $r>0$ let $\mathbb D_r=\{z\in \mathbb C:|z|<r\}$ and let $\bar{ \mathbb D_r}$ be its ...
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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 ...
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1answer
48 views

Prove that the given function is bounded, but not continuous, on the the given region

Let , $$f(z)=\begin{cases}e^{(−1/z)} & \text{ if } z\not=0\\0 & \text{ if }z=0\end{cases}$$ prove that the function is bounded, but not continuous, on the half circle $0\le|z| \le1, ...
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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?
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0answers
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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 ...
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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}}$ $ ...
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1answer
37 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 ...
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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 ...
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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 ...
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2answers
39 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
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0answers
38 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 + ...
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2answers
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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 $ ...
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1answer
283 views

Help with rearranging equation to get real and imaginary parts..

I know this is so simple but my algebra is totally failing me.. I have the equation 1/1+2i and I want to extract the real and imaginary parts so I have it in the form.. Re+Im could someone just ...
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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 ...
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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
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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 ...
2
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5answers
83 views

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

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