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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1
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1answer
24 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
0
votes
1answer
48 views

2x2 matrix multiplication issue

Let $$f_w(z)=z+w=\begin{bmatrix}1 & w \\ 0 & 1\end{bmatrix}z$$ where $z$ is a complex number. Shouldn't this be $w$ when $z=0$? However when I do the multiplication I get ...
10
votes
1answer
80 views

Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
2
votes
1answer
75 views

Complex integration on NON-simple closed curve

Compute the following integral with the help of Cauchy's residue theorem. $$\int_C\cot z\,dz$$where , $C:z=4e^{4i\theta}$ , $-\pi\le \theta\le\pi$ Here , singularities of are given by $\sin ...
2
votes
1answer
30 views

Are all interior points limit points in complex analysis?

The definition of limit point z for a set S in complex analysis states that there exists at least one point of the set inside the deleted neighbourhood of z.Does this imply that all interior points of ...
1
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1answer
42 views

Functions of complex numbers

Let $f(z) = \sqrt z$. Using the branch that is defined everywhere except where $z = x + iy$ with $y = 0$ and $x < 0$. What are the formulas for real valued functions $u(x, y)$ and $v(x, y)$ such ...
2
votes
1answer
155 views

Equality of complex numbers

I'm currently reading some notes on Complex Numbers and came across this 'proof' regarding the equality of complex numbers. Claim: Two complex numbers $a+bi$ and $c+di$ are equal iff $a=b$ and $c=d$, ...
0
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1answer
33 views

What are the real and imaginary parts of this complex propagation constant?

I am currently looking at the propagation constant $\gamma\in\mathbb{C}$, which is $$ \gamma = i\omega\sqrt{\mu\epsilon-i\,\frac{\sigma\mu}{\omega}}, $$ where $i^2 = -1$ and all other quantities are ...
0
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2answers
49 views

How to find the real or imaginary part of an equation involving complex numbers?

I am currently using the Debye model and need to find the real and imaginary parts of the equation. The Debye equation is $$ \epsilon_\text{r} = \epsilon_\infty + \frac{\epsilon_\text{s} - ...
14
votes
2answers
848 views

Prove that a polynomial has at least one nonreal complex root

Prove that the polynomial below has at least one nonreal complex root $$x^5+\frac{x^4}2+ \frac{x^3}3+\frac{x^2}4+\frac x{24}+\frac 1{120}$$ I have tried to prove that there exist $k\in \Bbb R$, such ...
1
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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
27 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
100 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?
1
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0answers
16 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?
0
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1answer
23 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
25 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 ...
1
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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
30 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
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3answers
64 views

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

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 $$(w_2-w_1 )(w_3-w_1 )\cdots(w_n-w_1 )=\frac n{w_1}?$$
0
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0answers
14 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
votes
0answers
77 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
42 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
420 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
votes
1answer
56 views

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

$Z_1,Z_2$ 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: ...
8
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9answers
185 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
votes
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
votes
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
49 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
votes
5answers
233 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
vote
6answers
76 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
vote
2answers
27 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
vote
4answers
70 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 ...
2
votes
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
vote
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} ...
1
vote
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
55 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: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$? ...