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
27 views

A question about complex using geometric.

Let $z_{1}$, $z_{2}$, and $z_{3}$ be three distinct complex numbers. Prove that these numbers are collinear if and only if the quotient $(z_{3}-z_{1})$ \ $(z_{2}-z_{1})$ is a real number. I have been ...
0
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4answers
47 views

Prove that $az^n+b\overline{z}^n=0$ does not have any complex solutions except for $0$ [closed]

Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$? The first one seems very obvious, but ...
3
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7answers
140 views

What does $\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3} $ imply?

I'm having trouble understanding what the following equality implies. $$\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3}.$$ I suspect that this means that the points form the vertices of an ...
2
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1answer
36 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
1
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2answers
56 views

Complex differentiability of $f(z)=|z|$

Why is the absolute value function $f : \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|$ not complex differentiable at any point $z_0$ in the plane?
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2answers
49 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
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1answer
62 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
3
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2answers
54 views

Square Rooting Back To Real Dimension

As we all know, square rooting -1 (a real number) opens up the "imaginary" dimension (defined by the presence of iota). We can return from the imaginary dimension back to the real dimension by ...
1
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3answers
56 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
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2answers
202 views

What is wrong with my proof: $-1 = 1$? [duplicate]

I have some theories about why this could by wrong but I still haven't something that convinces me. What is wrong with this proof: $ -1 = i^2 = i.i = \sqrt{-1}.\sqrt{-1} = \sqrt{(-1).(-1)}= \sqrt1 = ...
2
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3answers
154 views

Confused with imaginary calculus

So $i$ is the complex unit and $n \in \mathbb{N} $. $$e^{2 \pi \ n \ i} = 1$$ $$1^{2 \pi \ n \ i} = 1$$ $$(e^{2 \pi \ n \ i})^{2 \pi \ n \ i} = e^{-4\pi^2 \ n^2}$$ $$e^{-4\pi^2 \ n^2} \neq 1$$ I’m ...
4
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0answers
72 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
0
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1answer
35 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
2
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1answer
44 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
2
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1answer
34 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
0
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0answers
53 views

proof of a vector identity

In an exercise I am asked to prove the following vector identity: $$\textbf{a}\cdot\textbf{b}=\frac{1}{4}\big(|\textbf{a}+\textbf{b}|^{2}-|\textbf{a}-\textbf{b}|^{2}\big)$$ Both the dimension of the ...
1
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1answer
123 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
8
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2answers
319 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
1
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1answer
89 views

stuck with a cubic equation

I picked an equation $0= x^3 +x^2 -2x -1$ I plotted it with geogebra, to see if it had more than $1$ real root. It definitely cuts the $x$-axis $3$ times. But when I checked wolfram alpha, to see ...
5
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1answer
85 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
2
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1answer
25 views

Seperating points in the complex plane

Given a finite set of points say $p_1,p_2, \ldots, p_n$ in the complex plane, how do I find another point $q$ such that ray $R_i$ joining $q$ to $p_i$ are all distinct. I would be happy with any kind ...
3
votes
1answer
159 views

Limits of sequences connected with real and complex exponential

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to ...
0
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1answer
103 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
1
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0answers
47 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
3
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2answers
85 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
12
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4answers
375 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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2answers
43 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
1
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1answer
28 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
0
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1answer
43 views

Chain of inequlities in Complex variables

I am having difficulty understanding the following inequalities which is part of a solution to a problem: Suppose \begin{align} |z-1| &< 1/2\\ |z+1|&< 5/2\\ |z|&> 1/2\\ ...
2
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2answers
60 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
2
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0answers
121 views

Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
0
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2answers
621 views

Lagrange's identity in the complex form

I am trying to show Lagrange's identity in the complex form; that is, $$ \Bigl\lvert\sum_{i = 1}^na_ib_i\Bigr\rvert^2 = \sum_{i = 1}^n\lvert a_i\rvert^2\sum_{i = 1}^n\lvert b_i\rvert^2 - \sum_{1\leq ...
25
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7answers
6k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
2
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1answer
328 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
3
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0answers
68 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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1answer
64 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
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1answer
194 views

Distribution of magnitude squared for complex Gaussian

$\def\Re{\operatorname{Re}}\def\Im{\operatorname{Im}}$ If we have a random complex variable $h_l$, with $\Re[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ and $\Im[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ ...
0
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3answers
53 views

Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$

Let $z=\cos\theta+i\sin\theta$. Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$ Can anyone show me how to show the equation? I can't think of how to get $\frac 12 ...
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0answers
50 views

Question about galois imaginary and modular arithmetic

Let $p$ be a prime of type $3\space mod \space 4$. Then there is no solution $x^2 = -1 \space mod \space p$. Therefore we can define the so-called Galois imaginary $i$. ( $i^2 = -1 \space mod \space ...
0
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1answer
57 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...
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1answer
55 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...
1
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1answer
87 views

Complex Plane ( $\arg(z)$)

Sketch the following regions of the complex plane. For each, say whether it is open, closed, or neither, and whether it is connected. No proofs necessary. $$\left\{z \in \mathbb{C}\mid -\dfrac{\pi}{2} ...
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0answers
30 views

If two solutions of arg$(z)$ are in interval $−\pi<$arg$(z)≤\pi$ are both correct?

For example there is complex number $z=\sqrt3-i$ Are the answers $\frac{5}{6}\pi$ and $-\frac{\pi}{6}$ correct as $\text{arg}(z)$?
1
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1answer
47 views

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the polynomials

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the two polynomials $p(x)=5ix^4-(9+2i)x^3+7x+6-i$ and $q(x)=9x^5-x^3+7x+6$. The roots, with accurate to $10$ ...
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1answer
91 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
0
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2answers
46 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
0
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1answer
39 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
0
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1answer
45 views

Question about determining accumulation points

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not. a) ...
0
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2answers
59 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
1
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1answer
98 views

Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction). ...