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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2answers
22 views

Raising a number in Rectangular Form

What is the value of $(-2 + 3i\sqrt3)^6$? Answer is $4096$ Convert $(-2 + 3i\sqrt3)^6$ to Polar Form. $${ (\sqrt{31} \angle 111.05)^6 }$$ I use something called De Moivre's Theorem $${z^n = r^n( ...
2
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1answer
20 views

Question about asymptotic behaviour of argument of complex number

Let $r\in\mathbb{R}^{+}$, $\theta\in\mathbb{R}$ and $z_{0}\in\mathbb{C}$. Does $\arg{(r\text{e}^{i\theta}+z_{0})}\longrightarrow\theta$ as $r\longrightarrow\infty$?
0
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2answers
94 views

$i^i$ is real number. But $\ln(i^i)=i\cdot \ln(i)=\frac{i}{2}\ln(-1)$. But $\ln(-1)$ is not defined.

$i^i$ is a real number. But, $\ln(i^i)=i\cdot\ln(i)=\frac{i}{2}\ln(-1)$. But $\ln(-1)$ is not defined. So how can $i^i$ be a real number?
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0answers
15 views

Simplifying complex functions and expressions with real results

So I integrated a real function $$ \int_{0}^{k_{max}}\frac{k^4}{(k^2 + x)^2 + y^2} $$ $$= k_{max} + \frac{1}{2y} \left(i (x + iy)^{3/2} \arctan{\left(\frac{k_{max}}{(\sqrt{(x + i y})}\right)} - i (x ...
0
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0answers
17 views

Complex Normal Gaussian noise

I would like to create complex normal Gaussian noise with dimensions $(M,N)$ The noise should have zero mean and $var=1$. How can I do so?
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1answer
22 views

Complex number identity by trigonometry

Show that $\lvert e^{i\theta} - 1\rvert = 2\lvert\sin(\theta/2)\rvert$ by using the geometry of the triangle with vertices 0, 1, and the midpoint of the line joining 0 and $e^{i\theta}$. I have been ...
1
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2answers
24 views

A triangle and its median in complex plane.

Let $z_1$, $z_2$, $z_3$ be vertices of the triangle $\triangle ABC$. And given that $|z_1|=|z_2|=|z_3|$. Then the median through $A$ cuts the circumcircle at which point? We need to get the answer in ...
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2answers
145 views

Why are the roots of the polynomial $z^N = a^N$ equal to $z_k = a \ e^{j\frac{2 \pi k}{N}}$?

I am trying to understand equation 3.28 from this image in my book. I get everything that the author is saying, except for when he finds the roots, (zeros), of $z^N = a^N$. Of course, there are ...
1
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2answers
40 views

The magnitude of the complex number

How can we find $|M|^2$ for $$M=\frac{e^{2(1+{\rm i})l}-e^{-2(1+{\rm i})l}}{e^{2(1+{\rm i})x}-e^{-2(1+{\rm i})x}} ?$$ We have $$M=\frac{\cos 2l[e^{2l}-e^{-2l}]+{\rm i} \sin 2l[e^{2l}+e^{-2l}]}{\cos ...
1
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1answer
1k views

Straight Line Equation in Complex Plane

Hi there, I'm confused about the straight line equation in complex plane: how does "0 = Re((m+i)z + b)" come from "y = mx + b" ? I mean when I see "y = mx + b", I can draw a graph in my mind, but ...
1
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0answers
27 views

Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in ...
2
votes
3answers
63 views

Product of the difference of $n$th roots of $-1$ [on hold]

If $w_1,w_2,\ldots,w_n$ are the $n^{\text{th}}$ roots of $-1$, then how can we prove that by mathematical induction $$(w_2-w_1 )(w_3-w_1 )\cdots(w_n-w_1 )=\frac n{w_1}?$$
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0answers
13 views

Simplifying $\sum\limits_{n=0}^N -|a_n|^2+a_na_{n+1}^\star$

Can the sum mentioned above (where we set $N+1\equiv 0$ so that the sum is cyclic) be transformed to the form $\sum\limits_{n=0}^N -|\xi_n|^2$, where $\xi_n$ are linear combinatiosn of $a_n$?
3
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2answers
39 views

Factorisation over $\Bbb C$ of $z^2 -10z+30$

I haven't done these questions in a long time, so I am just wondering if my approach and answer is correct. When asked to $z^2-10z+30$ over $\Bbb C$, My approach: I complete the square of the ...
1
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2answers
37 views

complex numbers- proving the equality part in the Cauchy–Schwarz inequality using Lagrange identity

I need to discuss the equality case of: $$ \left | \sum_{k=1}^{n} z_{k}w_{k} \right |^{2} \leq \left ( \sum_{k=1}^{n}\left | z_{k} \right |^{2}\right )\left ( \sum_{k=1}^{n}\left | w_{k} \right ...
-4
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0answers
74 views

Find the cube roots of $-11-2i$.

How do I find the roots of $\sqrt[3]{ - 11 - 2i}$ ? Tried to use Moivre's theorem, but can not find the solutions by using the polar form: ...
2
votes
2answers
40 views

Complex exponential to real

I'm not yet very good at complex number, so I would appreciate the following insight: How exactly do we arrive from $e^{\pi(1-i)}-e^{-\pi(1-i)}$ to $e^{-π}-e^π$, and why does ...
0
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2answers
416 views

Find the cube roots of $ -8 i $ and plot them on a plane.

I can’t figure out the angle of this equation. I set it up like this: $$ z^{3} = 0 - 8 i. $$ I find that the $ r $-value is $ 2 $, but when I try to find the angle, I’m stuck. I can’t divide by $ 0 ...
0
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1answer
56 views

Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$ [on hold]

$Z_1,Z_2$ and $Z_3$ are affixes of points of equilateral triangle $ M_1 ,M_2$ and $M_3$. Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$.
4
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2answers
49 views

Why are values greater than $\pi$ radians given as negative in exponential form?

Find the fifth roots of $-3+3i$ in exponential form. My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$ Wolfram ...
1
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1answer
25 views

Multiplying square roots of negative numbers

I am just learning more about complex numbers and a question popped up I can't figure out on my own, so I've posted it here. I already know $i^2=-1$ and $i=\sqrt{-1}$ (isn't it even true that $\pm ...
7
votes
1answer
452 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
3
votes
1answer
81 views

Triangle inequality- complex

I am trying to prove the triangle inequality purely algebraically. Let $z=x+iy$, $w=u+iv$. Then, $|z+w|^2$=$|(x+u)+i(y+v)|^2$=$(x+u)^2+(y+v)^2$=$x^2+2xu+u^2+y^2+2yv+v^2$ I tried the other way: ...
9
votes
9answers
182 views

How to solve $z^3 + \overline z = 0$ [duplicate]

I need to solve this: $$z^3 + \overline z = 0$$ how should I manage the 0? I know that a complex number is in this form: ...
0
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0answers
38 views

matching the powers of the coefficients of polynomials

Hi: The result of the following question is stated (as an "it is straightforward to show that" type of result) in an econometrics paper, the link of which I can provide. But I translated into a ...
9
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1answer
150 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
2
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2answers
48 views

Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side ...
56
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25answers
7k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
1
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4answers
61 views

The real part of a complex number such that $|z|=\max\{|z-2|,|z+2|\}$

If |z|=max{|z-2|,|z+2|} then - INFERRENCE - |Re(z)|=1 Is the inferrence incorrect? My approach is - |z|=|z-2| when |z-2| {i.e. distance of z from 2 is greater}is greater OR |z+2| when |z+2| is ...
2
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5answers
229 views

Definitions for complex numbers

I could not find this question anywhere else. But why are addition, subtraction, division, and other operations defines they are in complex numbers? Could they defined as something else?
4
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3answers
89 views

The imaginary unit, $i,$ and an alternate representation.

Recently, I began working with both complex, and imaginary numbers, and I looked at the complex number $i^{n}.$ If $n = 0, i^{n} = 1,$ $n = 1, i^{n} = i = \sqrt{-1},$ $n = 2, i^{n} = i^{2} = i ...
0
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2answers
40 views

Identity with complex numbers related to the Cauchy-Schwarz inequality

I have this equation $ a_j,b_j\in \mathbb{C} , j=1,2,...,n$ $$ \left| \sum\limits_{j=1}^n a_jb_j \right|^2 = \sum\limits_{j=1}^n |a_j|^2 \sum\limits_{j=1}^n |b_j|^2 -\sum_{1\leq i \leq j \leq n} ...
1
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6answers
70 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
1
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2answers
26 views

Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
1
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4answers
69 views

Sum of roots of unity, proving that $1+w+w^2…+w^{n-1}=0$ [closed]

If $w$ is a unit square of rank $n$ (meaning $w^n=1$), s.t $w$ is not $1$. Prove that $1+w+w^2.....+w^{n-1}=0$. We're pretty sure that we need to use induction, its easy to prove for $n=2$ but ...
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0answers
58 views
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2answers
29 views

Fixed points of $\frac{1\pm \sqrt{1-|a|^2}}{\bar a}.$

Prove that $\phi_a(z)=\frac{a-z}{1-\bar az}$ , $0<|a|<1$ has exactly two fixed points ; one inside the unit disc and the other outside the unit disc. Putting $\phi_a(z)=z$ I find that there ...
1
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1answer
29 views

How can I visualize the interaction of the imaginary parts of the cosine/sine functions?

So I've been trying to get a good and intuitive feel for the extension of the sine and cosine functions into complex numbers (i.e. $\cos(z)$ where $z=a+bi$), and to do so I've naturally been looking ...
4
votes
0answers
59 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
-4
votes
0answers
27 views

set of points of orthogonal vectors [closed]

To every point M of affix Z we associate the point P of affix Z-1 and the point Q of affix z. Determine the set of points of M such that PM is orthogonal to pQ .
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2answers
75 views

Question about a step in the proof of the Cauchy-Schwarz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwarz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le \sum_{j=1}^n\vert ...
0
votes
4answers
53 views

How find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$?

Let $k > 1$ be a real number. How may one find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$? ...
1
vote
0answers
32 views

Complex numbers $x$, $y$, and $z$ are collinear, show that there exist $a,b,c$ for which $ax+by+cz=0$ where $a+b+c=0$

Suppose $x$, $y$, and $z$ are collinear (complex numbers). Prove that there exist real constants $a$, $b$, $c$, not all zero, such that $ax+by+cz=0$ where $a+b+c=0$. This is how I did it: First, ...
4
votes
2answers
88 views

What is the solution(s) to $1^i$?

$1^x$ is always $1$ with real numbers, but everything gets more complicated with complex numbers. Using Eulers formula, you know that $$e^{ix}=\cos(x)+i\sin(x)$$ If you make x=2π into this you'd get ...
0
votes
2answers
427 views

Find a solution to any single-variable equation

I know it is not possible to solve any equation of fifth degree and higher "using only a finite combination of the arithmetic operations and radicals in terms of the coefficients" (see on Wikipedia). ...
8
votes
8answers
825 views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
-1
votes
3answers
71 views

$ x+y = 1 $ and $ \frac{1}{x} + \frac{1}{y} = 1 $ Solve $ x^3 + y^3 $ [closed]

$x$, $y$ are complex numbers, $x$ and $y$ aren't $0$. $$ x + y = 1 $$ $$ \frac{1}{x} + \frac{1}{y} = 1 $$ $$ x^3 + y^3 = ? $$ Thank You!
3
votes
1answer
98 views

Finding an analytic function satisfying given two conditions.

Does there exists an analytic function $f:D\to D$ such that $f(1/2)=1/2$ and $f'(1/2)=-1$ ? If exists then find such a function. where , $D=\{z\in \mathbb C:|z|<1\}.$ I found that such a ...
1
vote
1answer
25 views

Checking whether points form a polygon in complex plane

If z^8=(z-1)^8 then the roots are 1) concyclic 2) form a polygonal 3)none I found the roots to be 1+cot(k.pi/8) for k is a natural number and less than 8. Then couldn't figure it out.
2
votes
3answers
61 views

Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

Let , $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$ denotes the semicircle traversed clockwise. Show that $$\lim_{r\to 0}\int_{C_r}\frac{e^{iz}}{z(z^2+1)}\,dz=-\pi i$$ I can not use the Jordan's ...