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

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2answers
41 views

Sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1 \}$

My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. I substituted $z$ for $a+bi$, but did not get anywhere: ...
0
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0answers
21 views

Is alpha and endomorphism of C considered as avector space over R? Is it an endomorphism of C considered as avector space over itself?

Let alpha:C$\to$C be the function defined by alpha:a+bi$\to$ -b+ai.(1) Is alpha and endomorphism of C considered as a vector space over R?(2) Is it an endomorphism of C considered as a vector space ...
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0answers
27 views

Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One ...
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0answers
29 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
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2answers
50 views

Finding the modulus of a complex number that satisfies a polynomial relation

Consider $z\in\mathbb{C}$ such that $z^2-2z+3=0.$ Find the modulus of $$f(z)=z^{17}-z^{15}+6z^{14}+3z^2-5z+9$$ My attempt: $z^2-2z+3=0\Leftrightarrow\left[ ...
2
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1answer
31 views

Cross ratio and symmetric points exercise

Problem Let $C$ be a circle or a line belonging to $\overline{\mathbb C}$ and let $z_2,z_3,z_4$. Two points $z$ and $z^*$ are said to be symmetric with respecto to $C$ if ...
4
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2answers
38 views

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ I went wrong somewhere, this is what I have so far (this is in polar): ...
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1answer
36 views

Möbius transformations on $\space \overline{\mathbb R}$

Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients. If it can be written with ...
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2answers
50 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
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0answers
12 views

what is deformation of a contour and Fourier inversion in general?

I would like to to know how one can do contour deformation and Fourier inversion in general? I shall be very thankful to you if some one explain it with the help of example.
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2answers
61 views

Solve for x if $z$ is a complex number such that $z^2+z+1=0$

I was given a task to solve this equation for $x$: $$\frac{x-1}{x+1}=z\frac{1+i}{1-i}$$ for a complex number $z$ such that $z^2+z+1=0$. Solving this for $x$ is trivial but simplifying solution ...
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3answers
30 views

Related to the construction of $\Bbb C$ (generalisation)

To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$ then write ...
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2answers
50 views

Trigonometric equation with complex numbers

Let $x$, $y$, and $z$ be real numbers such that $\cos x+\cos y+\cos z=\sin x+\sin y+\sin z=0$. Prove that $\cos 2x+\cos 2y+\cos 2z=\sin 2x+\sin 2y+\sin 2z=0$. Starting with the given equation, I got ...
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4answers
65 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
3
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1answer
38 views

Möbius transformation: proving the image of the unit circle is a line

Problem 1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively. 2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$. ...
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2answers
98 views

Why is this wrong (complex numbers and proving 1=-1)?

$$(e^{2πi})^{1/2}=1^{1/2}$$$$(e^{πi})=1$$ $$-1=1$$ I think it is due to not taking the principle value but please can someone explain why this is wrong in detial, thanks.
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0answers
25 views

Other complex systems

My question would be very short. As we all know, there are complex, quaternion number systems, which are based on multiplication and roots. So, my question is... Is there any other complex number ...
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1answer
19 views

Solving simultaneous equations with complex coefficients using real methods

My circuits analysis textbook teases that there's a way to convert a set of n complex equations into a set of 2n real equations, which can then be solved using any calculator that can solve real ...
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1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
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1answer
12 views

Images of the stereographic projection's inverse

I am trying to solve a problem which states: Let $\phi: \bar{\mathbb C} \to S^2$ be the inverse function of the stereographic projection Calculate $\phi(Re(z))=0$ and $\phi(Im(z))=0$. I can guess ...
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0answers
44 views

Complex numbers quesiton

z1 z2 z3 are three points on the gauss plane and create a triangle ; it is known that z1 = 3sqrt3 +3i, z2 = z1/cis240 deg , z3 = z2cis60 deg . given this data show that the triange is a staright ...
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1answer
42 views

Real part of Complex Function

I've this function $$f(k,\theta) = \frac{1}{k}\frac{1}{\cot\delta_0(k) -i }$$ and i know that $k\cot\delta_0(k) = -\frac{1}{a} + \frac{1}{2}r_ek^2 + \cdots$ it is an expansion. How can i get that ...
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0answers
35 views

Maximum of $P$ in the disk $|z|=1$ depending on co-efficients

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that ...
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3answers
85 views

What is the real and imaginary parts for the complex function $f(z)=z^z$

I know: $f(z)=z^z =|z|^ze^{iz\theta} $ and $=|z|^z(\cos(zθ) + i\sin(zθ))$ But how do I continue to get the results for $\Re(z^z)$ and $\Im(z^z)$? $$\text { }$$ Thanks.
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0answers
26 views

Complex roots of a complex number [duplicate]

I know how to find the roots to the equation $z^n=w$, for $n \in \mathbb{R\setminus\{ 0\}}$ (by writing $w$ as $re^{i(\theta+2k\pi)}$), and taking the nth root of both sides, which I'm perfectly happy ...
1
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2answers
76 views

Operations with complex numbers to give real numbers

If: $|z|=|w|=1$ $1 + zw \neq 0$ Then $\dfrac{z+w}{1+zw}$ is real. How can prove that.
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3answers
29 views

Product of the n roots of the unity can be $(-1)^{n-1}$

I need to prove that the product of the n roots of the unity can be $(-1)^{n-1}$. If i make $1^n=z$, z can be $ cis(\dfrac{k \cdot 2\Pi}{n})$ with k=0, ... , n-1 Now i need to prove that ...
4
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1answer
45 views

relationship between complex numbers

Consider the following: Two equilateral triangles inscribed in a circle. The vertices of the large triangle are the geometric images of the three cubic roots of $z$ (a complex number). The small ...
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1answer
28 views

Principal values of complex functions

How do I find the principal value of the following: $\log(1-\sqrt{2i})$ And hence of $(1-\sqrt{2i})^i$ Also how do I write $z=1+i$ in polar form and find its roots? I find these very confusing ...
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1answer
38 views

Proving the Mandelbrot set is bounded

I am trying to prove that the Mandelbrot set defined as the set $\mathcal M$ of complex numbers $c$: the recursive sequence defined as $$z_0=c, \space \space \space z_{n+1}={z_n}^2+c$$ is bounded. ...
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1answer
23 views

Cauchy-Riemman $w = |z^2|$

So for these types of questions, I can compute the partial differentials for Cauchy-Riemann but then I have trouble seeing/explaining where the function is differentiable? For example with this ...
6
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1answer
63 views

Prove that $\cos(z)$ and $\sin(z)$ are surjective over the complex numbers. [duplicate]

I have an exercise that says: (a) Prove that $\cos(z)$ and $\sin(z)$ are surjective functions from $\mathbb C \to \mathbb C$. (b) Find the solutions of the equation $\cos(z)=\dfrac{5}{4}$. As far ...
2
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4answers
591 views

Is the power of complex number defined yet?

Let $z$, and $b$ be two complex numbers. What is $$f_b(z)=z^b.$$ If I write it like this: $$ \left(re^{i\theta}\right)^{b}=r^{b}e^{ib\theta}. $$ Would this even make sense? Wolframalpha gives me ...
0
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1answer
77 views

How to find the roots of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ using trigonometric methods

Can all the roots (real or complex) of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ be found using trigonometric methods? Many thanks to all of answers.
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2answers
54 views

How to prove that :$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*$

Can someone tell me how to prove the folowing equalty : $$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*.$$ Thanks in advance.
0
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1answer
37 views

Determining Complex number

Let $ z_1,z_2$ $\in$ $\mathbb{C}$ and A,D their respective images in the complex plane. let B be the image of $z_1³$ in the complex plane. A is in the first quadrant and B is in the second quadrant. ...
0
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1answer
38 views

Proving $\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$

Prove that $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}=\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$$ I tried to use $$ ...
2
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4answers
91 views

Proving by calculation that $\arg(-2) = \pi$

The fact that it is true, seems very obvious, if one draws the complex number $z = (-2 + 0i)$ on the complex plane. The angle is certainly 180 degrees, or pi radians. But how can this be proven by ...
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2answers
46 views

Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

How to write: $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in $$(a+i b) $$ $$ ?$$ I tried to multiplicate by $$e^{i}$$ (the numerator and ...
4
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4answers
56 views

Proof of trigonometric identity $\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$

I was given a task of proving the following identity: $$\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$$ I am not looking for a solution, just some kind of a hint to start ...
0
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1answer
16 views

Complex number Q1

For $z\in \mathbb{C}\backslash \{i\}$. How can i go from this line : $|z|^2-Re[(1-i)z]=0$ To that one : $$\left|z-\dfrac{1+i}{2}\right|^2=\left|\dfrac{1+i}{2}\right|^2=\dfrac{1}{2}$$ indeed, ...
0
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1answer
45 views

If Re(z) >0, then the value of |1+ z+ z^2…z^n| cannot be less than?

Here are the options- a)$|z|^n - 1/|z|$ b) $|z|^n + 1/|z|$ c) $n|z|^n$ d) $n|z|^n + 1$ The answer is supposed to be $|z|^n - 1/|z|.$ How do I solve this question? I tried writing $1+ z+ z^2...z^n$ ...
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4answers
65 views

Finding all roots of $z^4-4z^3+9z^2-4z+8$

I need to know all the roots of $z^4-4z^3+9z^2-4z+8$. I know only one root: z=i. Is there an easy way to find the 3 roots that are unknown? thanks.
2
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0answers
58 views

How the total order property of $\mathbb{R}$ is related to not being algebraicaly closed?

The field of real numbers $\mathbb{R}$ is total-ordered and not algebraicaly closed, the field of complex numbers $\mathbb{C}$ is not ordered but is algebraicaly closed. Intuitively how these two ...
1
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1answer
38 views

complex equations question

find all solutions of the equation: $w^4 = -8(1-i\sqrt{3})$ I dont wanna be that guy, but can someone tell me what the second solution to this equation is? cuz the solution manual says it's $-1 + ...
2
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2answers
53 views

Inequality in complex numbers

Prove that for all $z\in \mathbb{C}$ $$\frac{\Vert z+i\Vert z\Vert \Vert}{\Vert z+1\Vert}\leq \frac{2\Vert z \Vert}{\Vert z \Vert +1}$$
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3answers
35 views

Find the principle value and all other values of $i^{2/\pi}$

I'm a little confused about how to go about this? Any help would be appreciated, thanks.
1
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1answer
64 views

Using Newton's method to solve a non-linear system of equations over complex numbers

I have a function $f(\bar{z},z)$ mapping from $\mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}^n$, which I would like to find the roots of numerically. Since it is nicely formulated in terms ...
1
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2answers
37 views

$z=(-1+i)^{11}+(-1-i)^{15}=?$

Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so $z=-96+160i$ $z=96-160 i$ $z=160-96i$ $z=-160+96i$ what is the right answer ? Thanks in advance.
1
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2answers
91 views

How to separate out real and imaginary terms

I have an equation like this: $$a+ib = \log(x+iy).$$ I need to separate the real and imaginary part in RHS so that I can equate the real part of LHS to real part of RHS and imaginary to imaginary ...