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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5answers
67 views
+150

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 ...
0
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2answers
83 views

If $f$ is an entire function then what about the set $S=\{Ref(z)+Imf(z) :z\in D\}.$

Let, $f$ be an entire function on $\mathbb C$ and let $D$ be a bounded open subset of $\mathbb C$. Let, $$S=\{Ref(z)+Imf(z) :z\in D\}.$$ Which of the following(s) is(/are) necessarily true ? (a) $S$ ...
3
votes
4answers
73 views

$z^3=w^3 \implies z=w$?

I've reached this in another problem I have to solve: $z,w \in \Bbb {C}$. $z^3=w^3 \implies z=w$? I've scratched my head quite a bit, but I completely forgot how to do this, I don't know if this is ...
1
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2answers
35 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 ...
2
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2answers
57 views

Commutative binary operations on $\Bbb C$ that distribute over both multiplication and addition

Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition? In other words, if our operation is denoted by $\odot$, then I want the ...
1
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1answer
47 views

Projection of the XY plane.

Is there a way to project the infinite Complex plane to either the Poincare disk or the unit disk - for all values of x + iy ?
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0answers
45 views

Julia and Mandelbrot Sets [on hold]

I need to know how escape, prisoner, Julia and Mandelbrot sets work. Are they all in one sequence or are they separate.
4
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4answers
200 views

Proof: Derivative of $(-1)^{x}$

The derivative for $(-1)^{x}$ is \begin{equation} \frac d{dx}\left[(-1)^x\right]=i\pi(-1)^{x} \end{equation} But why? What happens with higher order derivatives? Thanks in advance.
0
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0answers
9 views

Statistical significance of deviation from a complex-valued model

I have complex-valued data. At each one of about 100 linearly spaced $x$ values, I have a corresponding measurement of a complex quantity with well-defined Gaussian uncertainties on both the real and ...
1
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1answer
62 views

How does one use the complex plane to solve this problem?

Given: $$a^2 + ab + b^2 = 1 + i$$ $$b^2 + bc + c^2 = -2$$ $$c^2 + ca + a^2 = 1$$ Find $$(ab + bc + ca)^2.$$ The solution says to use the complex plane. Can somebody explain to me (an average ...
0
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0answers
64 views

Complex numbers: conjugate [on hold]

Can anyone help me with this? Let $z^*$ be the complex conjugate of $z$ (a) Show that $ (zw)^* = z^*w^*$ (b) Prove by induction that $\ (z^n)^*=(z^*)^n$ for all positive integers $n$.
2
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1answer
46 views

Geometric series and complex numbers [duplicate]

I'm new to this site, english is not my mother tongue, and I'm just learning LaTeX. I'm basically a noob, so please be indulgent if I break any rule or habits. I'm stuck at proving the following ...
1
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3answers
65 views

Finding all the values of $\sqrt[3]{7-4i}$

I'm reading about De Moivre's Formula and the Roots of Unity, and one of the exercises is to find all the different values of $$ \sqrt[3]{7-4i} $$ I know that you can find the $n$th root of 1 with ...
1
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2answers
48 views

Finding all complex roots of this equation

So i have this equation: $z^5-4z^4+11z^3+12z^2-42z+52=0 \text{ for }z\in\Bbb{C}$ One root is: $z=1+i$ That gives us also the 2nd root. $z=1-i$ But i am stuck with how to get other 3. I thought i ...
0
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0answers
48 views

Is $i^i$a real number or not? [duplicate]

How might we go about proving $i^i$ is not a real number? I don't know in general how to exponentiate to complex powers, I found this question in an introductory calculus course, so maybe there is an ...
1
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1answer
35 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
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1answer
55 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 ...
11
votes
1answer
85 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
78 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
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1answer
33 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
159 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
36 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
56 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
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2answers
851 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 ...
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2answers
24 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
votes
1answer
39 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
103 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. [on hold]

$i^i$ 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
17 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?
1
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2answers
27 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
votes
3answers
65 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 $$(w_2-w_1 )(w_3-w_1 )\cdots(w_n-w_1 )=\frac n{w_1}?$$
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0answers
16 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
votes
2answers
43 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 ...
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0answers
86 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
43 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
426 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 ...
4
votes
2answers
50 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
454 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
189 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
39 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
votes
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 ...