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

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Show that for $z $ a complex number, there exists a complex number $\alpha $, wiht $|\alpha |=1$ such that $\alpha z = |z |$

How can I show that for $z $ a complex number, there exists a complex number $\alpha $, wiht $|\alpha |=1$ such that $\alpha z = |z |$ Thanks in advance!
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1answer
59 views

Find all the roots of this complex equation

Let $C$ be the set of complex numbers and $j$ the imaginary unit. Find all the roots(in $z$ $\in$ $C$) of the following equation: $$ 2z^7 + 6z^4 = z^3e^{-j{\frac π7}} + 3e^{-j{\frac π7}} $$ ...
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0answers
23 views

Random Sampling of vectors on the Complex Unit Sphere

This is my first post in these forums. Working in Mathematica, I would like to generate a large number (10000) of randomly sampled vectors on the complex unit sphere in n dimensions. I am not ...
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5answers
381 views

How do you find the imaginary roots of a fourth degree polynomial that cannot be simplified?

I started out with $f(x)=16x^6-1$, and I got it down to $64x^4+16x^2+4$ by synthetically dividing by roots $0.5$ and $-0.5$ How should I continue in order to find the other roots?
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0answers
18 views

The real part of a linear function

I apologize before hand for any misuse of terminology as English is not my first language, if in doubt about anything please do ask. In a subspace $W\subset(C^\infty(\mathbb{R}),\mathbb{C})$ spanned ...
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2answers
24 views

Solve equation with complex numbers using a helper equation

For the last two hours I've been trying to solve this complex equation using a helper equation. But I can't work it out. $z^2 = 5-12$ $\text{Let} \space z = x + yi$ $(x+yi)^2 = 5-12i$ $x^2-y^2 + ...
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0answers
28 views

Points on a unit circle

Let $P_1, P_2,..., P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2,3,...,2013\}$ is the product of all pairwise distances: $$\prod_{1\le i\lt j\le n} P_{i}P_{j}$$ a ...
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0answers
29 views

Laurent Series Expansion of $e^{z+\frac{1}{z}}$ about $z_0=0$

As the question title states, I'm tasked with finding the Laurent series expansion for $e^{z+\frac{1}{z}}$ about $z_0=0$. My approach is as follows. $$ \begin{align*} ...
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0answers
29 views

How to find complex roots using Müller's Method?

I have $f(x) = 2x^5 - 2x^4 + 6x^3 - 6x^2 + 8x - 8$ and $x_0 = 0.4$, $x_1 = 0.6$, $x_2 = 0.5$ with a tolerance Es = 10^-4. I solved it using Müller's method in 4 ...
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4answers
102 views

How is $\ln(-1) = i\pi$?

How do I derive: $\ln(-1)=i\pi$ and $\ln(-x)=\ln(x)+i\pi$ for $x>0$ and $x \in\mathbb R$ Thanks for any and all help!
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3answers
91 views

How to define the complex square root $ \sqrt{z} $?

We need to define the complex square root $ \sqrt{z} $ on a small open $ U \subset \mathbb{C} $, for example a disc. Let put : $ \mathcal{F} (U) = \{\ f: U \to \mathbb {C} \ / \ f \ \text{is ...
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0answers
29 views

Laurent Series Expansion of $\dfrac{z^2}{z^2-1}$ about $z_0=1$

So, as the question states, I'm trying to find the Laurent series expansion of $\dfrac{z^2}{z^2-1}$ about $z_0=1$. I've tried fiddling with geometric series stuff (the form of the rational function ...
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1answer
16 views

Complex Analysis - Complex plane, differentiable

Determine all the points in the complex plane where the function f(z) = tan(z) is differentiable and calculate the derivative at those points.
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2answers
43 views

calculate for any natural number n $\cos(2\pi/(2n+1))+\cos(4\pi/(2n+1))+\cdots+\cos(2n\pi/(2n+1))$ [closed]

How to calculate for any natural number? $$\cos\bigg(\frac{2\pi}{2n+1}\bigg)+\cos\bigg(\frac{4\pi}{2n+1}\bigg)+\cdots+\cos\bigg(\frac{2n\pi}{2n+1}\bigg)$$
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2answers
93 views

Why does $1^{-i}$ equal 1? [duplicate]

At one point, I found an equation that works with complex logarithms, but I lost the book that contains the equation. If I feed this to Wolfram|Alpha, it states that $1^{-i}$ is equal to 1. Why is ...
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0answers
35 views

Complex number series

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me. I have been trying to crack the following questions, i hope you could shed the light ...
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1answer
40 views

Solve the equation $\lvert z\rvert^2z-3\overline z=0$

I was trying to solve the equation using the identities $z=x+iy$; $\overline z=x-iy$ and $\lvert z\rvert^2=x^2+y^2$ so as to get $(x^2+y^2)(x+iy)-3(x-iy)=0$, that is $x^3+x^2iy+xy^2+iy^3-3x+3iy=0$ ...
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1answer
23 views

Roots of Complex Equation

Find the 3rd Root of equation $z= 8\left(\cos(\frac \pi 4) + i \sin(\frac \pi 4)\right)$ If I write the values of cos and sin then I would have only one root, how to find 3 roots for above equation ? ...
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2answers
40 views

How to divide a complex number by another complex number?

Is this true to say that : I don't find this definition in my book but it is the feeling that I had while looking at exercises correction.
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1answer
92 views

Study the character of the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$

Discuss the character of the series $$\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$$ where $z\in \mathbb C$ and $|z|=\frac{1}{4}$. Any suggestions please? Thank you very much
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2answers
62 views

Prove complex equation

Prove the following $$\frac{1}{z-1}*\frac{1}{z^n}= \dfrac{1}{z-1} - \sum_{k=1}^n\frac{1}{z^k}$$ for any integer n greater than 0. DO NOT USE .... I believe that I can use mathematical induction. ...
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1answer
51 views

Find the value of $\frac{w+1}{1-w}$ given that $w^2=-1$

Question There is a new real number $w$ such that $w^2 = -1$. If all the laws of arithmetic applies, find the value of $\dfrac{w+1}{1-w}$ . I tried the following: $$\frac{w+1}{1-w} = ...
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1answer
53 views

Factorial of $i$

With $i^2=-1$, what is $i!$ ? I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$ It would make sense either way. Is it not defined?
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1answer
30 views

Diagonalization with complex eigenvalues

Given $A = \begin{bmatrix} 0 & 1 \\ -4 & 0 \end{bmatrix}$ find a matrix C of the form $ C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and a matrix P such that $A = PCP^{-1}$. ...
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1answer
37 views

Minimizing a function of a complex variable

Given complex numbers $z_1,z_2,z_3,\ldots,z_n \in \mathbb{C}.$ Does there exist a $z \in \mathbb{C}$, for which the function $$f(z) = \sum_{j=1}^n |z-z_j|$$ achieves a global minimum? If yes, ...
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1answer
58 views

Given $x$ and $y$ in $\mathbb{Z}[i]$, find $q$ and $r$ such that $x=qy+r$.

Find $q, r \in \mathbb{Z}[i]$ such that: $1 + 5i = (1 + 2i)q + r$ with $|r| < 2$, $1 + 5i = (2i)q + r$ with $|r| < 2$. My only train of thought is that $r = 1+0i$, $0+i$ or ...
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1answer
38 views

. The equation $x^3 +10 x^2 − 100 x + 1729 = 0$ has at least one complex root $α$ such that $ | α | > 12$. [duplicate]

Show that the equation, $x^3+10x^2−100x+1729=0$ has at least one complex root $z$ such that $|z|>12$.
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3answers
36 views

Norm of Polar Coordinate in Unit Disk [closed]

Does $|e^{i\theta}|^2$ = 1 for $0<\theta<2\pi$ on the closed unit disk?
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1answer
15 views

Solve the given Functions for Complex numbers.(Complex Functions)

I tried solving for the real solutions and found myself stuck. Here is how i tried to solve it: Re(z)=1 f2(M2)= f2(1+iy) {z=1+iy} i'm not so sure about this. f2(m2)= (1+iy)^2 {as z^2 is f2} ...
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2answers
48 views

Primitive of an analytic function - Proof verification

Check out the proof for the following corollary. This is only the first part of the proof and I have a issue with this. $F(z)$ is defined by integrating $f$ along a line segment. But isn't it ...
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0answers
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'Simple' Vector analysis and I need a little help prooving an identity.

I have to prove that $$\vec{S}(\vec{r}) = -\frac{c^2}{\omega} \bigg( u(\vec{r})\vec{\nabla}\phi(\vec{r}) + \frac{i}{2}\vec{\nabla}u(\vec{r}) \bigg) ~~~~(1)$$ given that $$\vec{S}(\vec{r}) = ...
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3answers
27 views

Absolute value of z = the absolute value of z+3i, find the imaginary part of z.

A complex number is such that the absolute value of z = the absolute value of (z-3i) (a) Show that the imaginary part of z is 2/3. I tried squaring both sides and then I find by equating both sides, ...
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2answers
288 views

Can a set containing 0 be purely imaginary?

A purely imaginary number is one which contains no non-zero real component. If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? My issue here is that ...
3
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2answers
42 views

Complex numbers and geometric series

a) Use the formula for the sum of a geometric series to show that $$\sum _{k=1}^n\:\left(z+z^2+\cdots+z^k\right)=\frac{nz}{1-z}-\frac{z^2}{\left(1-z\right)^2}\left(1-z^n\right),\:z\ne 1$$ I thought ...
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1answer
64 views

De Moivre's use to a complex number

I have this question which I'm stuck on, here's the question and what I did. Find the smallest positive integer m such that $\left(\sqrt{3}+i\right)^m=\left(\sqrt{3}-i\right)^m$. I expanded out each ...
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6answers
67 views

Complex numbers equation with modulus and De Moivre's formula

I have a problem with the following question. For which $n$ does the following equation have solutions in complex numbers $$|z-(1+i)^n|=z $$ Progress so far. Let $z=a+bi$. Since modulus ...
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1answer
35 views

Find the principal value of $z=\ln\left(i\tanh\left(\frac{\pi}{2}\right)\right)$

Steps that I have taken: Substitute $\tanh(π/2)$ with $C$. Then we have $\ln(i\cdot C)=\ln|c| +i(\arg z)$. I am desperately stuck here. I also tried expressing $\tanh$ by definition using the powers ...
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1answer
17 views

Factoring completely using complex cube of unity

How can you completely factor $a^2 + ab + b^2$ and $a^2 - ab + b^2$ completely using $\omega$, the complex root of unity? Is there some general rule for such complex factorisations? Any help would be ...
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1answer
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Show that $-2, -1, 0, i$ lies in the Mandelbrot set but that $1$ lies outside of it

The Question Let $c$ be a complex number. The complex numbers $z_n(c)$ are defined recursively by $z_1(c)=c$, $z_{n++1}(c)=(z_n(c))^2+c$ for $n\geq1$ The Mandelbrot set is defined by ...
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3answers
320 views

A difficult inequality involving complex numbers

Suppose that $z_1,\ldots,z_n$ are complex numbers with the property that there is some constant $C$ such that $$\big|z_1^r+\cdots+z_n^r\big|\leqslant C$$ for all integers $r\geqslant0$. Show that ...
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2answers
78 views

How to show that $e^{-(-2)^\frac{2}{3}}$ is complex.

I am baffled with showing that $e^{-(-2)^\frac{2}{3}}$ is complex. My understanding is: $$ e^{-(-2)^\frac{2}{3}}= e^{-(4)^\frac{1}{3}}$$ since ${-(4)^\frac{1}{3}} $ is negative real number, so $ ...
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1answer
142 views

If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?

Let $A$ be a nonzero real number and let $B$ be a nonreal complex number. Let $z$ be a complex number. Let $f(z)$ and $g(z)$ be non-constant functions defined for all complex numbers $z$ and satisfy ...
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0answers
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Characterization of the complex unit ball?

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$. Take $U\subsetneq \mathbb{K}$ open neighborhood of zero verifying: $U=U^2 =\{uv\,:\,u,v\in U\}$; $U=-U$; $\mbox{int}(\mbox{cl}(U))=U$; $1\in ...
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2answers
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Solve the equation: $z^2 - (7 +6i)z + 4 +22i = 0 $ in $\mathbb{C}$

I am doing some repetition for fun and got stuck on this question: Solve the equation: $z^2 - (7 +6i)z + 4 +22i = 0 $ in $\mathbb{C}$ This is actually on the chapter of polynomials so I guess ...
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1answer
36 views

Multiplying two fractions with complex numbers

I'm doing $$ \frac{6-7i}{1+i}\cdot\frac{1+i}{1+i}, $$ and I'm getting the correct value for the numerator (namely, $-1-13i$), but based on the problem answer, I need for the denominator to become $2$. ...
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1answer
69 views

Plotting these complex inequalities on a complex plane.

For the first part a) it is very clear a circle with center at -i. With common sense using equation of a circle. For the 2nd part i'm having some trouble. My steps.. b) x+2y<3 $$(\sqrt{x})^2 ...
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2answers
81 views

The roots of a complex number in complex plane. [closed]

This question involves a very simple expression of a complex number to power 5. How can I quickly identify if the complex number has exact 5 different complex roots and may be some roots have the same ...
2
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1answer
29 views

How to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$

I am solving an exercise in my textbook. After some steps, I need to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$, where $\alpha\neq 0,0<\beta<1$, in order to ...
3
votes
3answers
66 views

Evaluate $\int t^2 e^{-2i\pi nt}\,dt$

I need to get $$\int t^2 e^{-2i\pi nt}\,dt$$ I'm thinking to use integration by parts, but $\int e^{-2i\pi nt}\,dt$ is tripping me up. Can anybody help? Thanks!
19
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2answers
2k views

Is it possible to solve for i? Is it infinite?

I was messing with the identity $e^{i\pi}=-1$ and I got that $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$ and on. I plugged it in to a calculator and it was infinite. It grew very fast. Does that ...