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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3answers
78 views

Can some one explain to me what is going on here - power of complex number

So here is the question and the work to solve it, but I have no idea how one knows to do the first step or what the first step is... $$ \begin{align} (6-i\sqrt{12})^{12} &= ...
1
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3answers
49 views

Finding three roots of a complex number if we already know one root

If we know that $a+bi$ is one of the forth roots of the complex number $z$, how can we find the other three roots?
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3answers
38 views

need help explaing the complex roots of a cubic

I am trying to understand a Galois theory example and we are looking at the solutions of $x^3-2=0$. It says they are $2^\frac{1}{3},2^\frac{1}{3}\omega, \text{ and } 2^\frac{1}{3}\omega^2$. I know ...
0
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2answers
36 views

How to solve these three complex-numbers equations

I'm so stuck right now studying for my bachelor of science math exam. Please show (using i = imaginary unit): 1) i^(1/i) = e^(PI/2+k*2*PI) 2) (4*i)^(1/2) = { 2^1/2*(1+i) ; 2^1/2*(-1-i) } 3) i^(i*pi) ...
0
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1answer
62 views

Proving that $f(z)$ is polynomial [duplicate]

Given $R>0 , M>0$ Let $f(z)$ be a entire function such as $|f(z)|\leq M|z|^m$ for evey z such as $|z|>R$. Show that $f(z)$ is a polynomial of degree m or lower.
1
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1answer
59 views

Prooving that a complex function is constant

Let $f(z)=u(z)+v(z)i$, $u(z) \leq 0 , \forall z \in \mathbb{C}$, and $f(z)$ is entire. Then $f(z)$ is constant. The hint is "use the Liouville theorem". I tried , but i need prove which f is limited ...
0
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1answer
63 views

Complex function and Jacobian matrix

Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so: $$\dfrac{\partial u}{\partial ...
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1answer
23 views

About the convergence or divergence? [closed]

Whether the following integral converge or diverge by comparison test. \begin{align*} ...
2
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3answers
50 views

How can one prove $||z_1|-|z_2||\le|z_1+z_2|$?

How can it be shown that for the complex numbers $z_1$ and $z_2$: $$||z_1|-|z_2||\le|z_1+z_2|$$ My text provides a hint that $z_1=z_1+z_2+(-z_2)$, and $z_2=z_1+(-z_1)+z_2$. $${}$$
1
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2answers
117 views

Expansion of $\sin^5 \theta$ using the Complex Exponential

How do I expand $\sin^5\theta$ using the complex exponential, in order to obtain: $$\frac{1}{16}\sin 5\theta - \frac{5}{16}\sin 3\theta + \frac{5}{8}\sin\theta$$ Thank you.
1
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4answers
71 views

$ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and no idea about $\sum\limits_{n=1}^\infty \dfrac{(3+4i)^n}{5^n\,\sqrt[999]{n}}$

How can one see that $ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and by which criterion? I was using a binomial theorem for $ (1+i)^n $ as $ \sum\limits_{k=0}^n \dbinom{n}{k} i^n$, ...
4
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6answers
506 views

Is $i$ irrational?

On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational. However, the set of irrational numbers, ...
0
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2answers
68 views

How to solve this equation? $8z^3+1=0.$ [closed]

I would like to know how to do this question. (a) Determine all of the solutions of the equation $$8z^3+1=0.$$ Express your solutions in the form $z = re^{i\theta}$ where $-\pi<\theta \leq \pi$. ...
-1
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1answer
70 views

Draw a set of values in complex plane where the complex number $w=1-3i$ is pure imaginary number.

How would you draw a set of values (in complex plane) where the complex number $w=1-3i$ is pure imaginary number? Could this be the solution? If $Rew=0$.
0
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1answer
166 views

How to solve the complex equation? $(x+2yi)^2 = xi.$

How to solve the following complex equation with in less than 60 seconds? $$(x+2yi)^2 = xi.$$ I know how to solve, we have to solve power first then real part equal to real part and imaginary to ...
0
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0answers
47 views

Schwarz's Lemma and extensions in complex analysis

I was assigned this problem: (which I here present verbatim) Let $f$ be a holomorphic function of the unit disk unto itself. Prove that $|f'(0)|\le 1$. Isn't it also necessary to assume that ...
0
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2answers
105 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
1
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2answers
82 views

Prove that $z_1, z_2, z_3, z_4$ are the vertices of a rectangle if and only if…

I have to prove that $z_1, z_2, z_3, z_4$, where $|z_1| = |z_2| = |z_3| = |z_4| = 1$, are the vertices of a rectangle if and only if $z_1z_2z_3+z_1z_2z_4+z_1z_3z_4+z_2z_3z_4=0$ Any help? There is a ...
1
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1answer
79 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
0
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1answer
50 views

Complex summation simplification

What I'm getting is $$\frac{( \sin (N+1)x - 2^N \sin x)}{(2^N(\sin x - 2))}$$ How do I simplify to the form they have given , please help. I hope it's clear because I don't know Ajax still ...
2
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1answer
43 views

Summation of complex numbers and simplification

By considering $$ \sum_{k=0}^{n-1}(1+i\tanθ)^k\tag{1}$$ Show that $$ \sum_{k=0}^{n-1}\cos(kθ)\sec^kθ=\cotθ\sin(nθ)\sec^nθ\tag{2}$$ Provided $θ$ is not an integer multiple of $\frac{π}{2}$. My take ...
3
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3answers
81 views

System of equations with complex numbers

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me. I am trying to find an efficient way to solve the following system of equations: $$ ...
2
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1answer
119 views

absolute value of sum of complex numbers squared

is this correct $ \left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2|a||b|+|b|^2$ Thank you
1
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1answer
149 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
1
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1answer
60 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
6
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4answers
550 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
-1
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2answers
84 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
1
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1answer
73 views

Find the real and imaginary parts of $\sin(\frac{\pi}{2}+i\ln2)$

Find the real and imaginary parts of $$\sin\left(\frac{\pi}{2}+i\ln2\right)$$ Using the double angle formula I have gotten ...
1
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1answer
75 views

Solve $(\frac{z+1}{z})^5 =1$ using fifth roots of unity

$$(\frac{z+1}{z})^5=1$$ Show that its roots are $$-\frac{1}{2}(1+i\cot(\frac{kπ}{5})), k = 1,2,3,4$$ I need to use the five fifth roots of unit, with angles $0,\frac{π}{5}, ...
22
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8answers
791 views

Is $e^{i\pi}+1=0$ all it's cracked up to be?

While it is beautiful and elegant and all that, isn't it true that Euler's identity is really just an artifact of how we define the radian? I'm speaking of those who say that it's great because it ...
1
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2answers
63 views

Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$

I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you. Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for ...
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2answers
92 views

What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
0
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1answer
124 views

Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$

This is an old qualifying exam problem that I am working on. I would appreciate some help. Thank you. Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$ and ...
0
votes
1answer
89 views

Proof using de Moivre's Theorem

Let $z=\cos\theta + i\sin\theta$ Show that $$1+z = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2})$$ I don't even know how to start on this proof.
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1answer
37 views

An inequality on the real part of a square root

I have the following inequality: $\Re(k+z) \geq \Re \sqrt{(k+z)^2-4z}$ where $k$ is real and $z$ complex. Under what conditions on $k$ and $z$ is this inequality true? I suspect that it is true for ...
0
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1answer
69 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
0
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1answer
68 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...
1
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1answer
53 views

Imaginary unit $i$ is not a limit of a real Cauchy sequence

I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued ...
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2answers
52 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
0
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1answer
43 views

Let z != -1, which module is 1. Prove that z is presentable z = (1+ti)/(1-ti), where t is real number

Let z != 1, which module is 1. Prove that z is presentable in the following form: $$ z =\begin{align} \frac{1 + ti}{1 - ti} \end{align}$$ where t is a real number So, im guessing i have to ...
0
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1answer
68 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
2
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1answer
106 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
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2answers
61 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
1
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1answer
104 views

This integral is strange

$$ \int_{C_1}\frac{dz}{z}=\int_0^{2\pi}\frac{-R\sin{t}+iR\cos{t}}{R\cos{t}+iR\sin{t}}dt=\int_0^{2\pi}i\text{ }dt=2\pi i\tag{24.36} $$ Shouldn't it simply be $$\left[\ln(R \cos t + iR \sin ...
0
votes
2answers
345 views

What is the polar form of -6i?

The module of -6i is 6 (the square root of 36), but $ tan\theta = -6/0$, meaning that the polar form $ 6(cos\theta + isen\theta) $ is also indefinite?
0
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2answers
29 views

Re z/z continuous at z=0

How would I show that Re z/z is continuous at z=0? I know that the real value of a complex number equals the sum of the real number and its conjugate divided by two, but I'm not sure where to go when ...
0
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2answers
67 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
1
vote
1answer
59 views

Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
2
votes
2answers
91 views

Given that $x$ is a rational number, is $\sin(x\pi)$ always expressible through radicals?

This is a theory I just thought of and I am wondering if there is truth to it. Here is the logic that I am working upon: Using Euler's formula, you can deduce that $$ (-1)^x = ...
0
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0answers
33 views

What is the name for the operation of swapping the two components of a complex number (rectangular form)?

I wonder if there is a name for the operation of swapping the real and imaginary part of a complex number.