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

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43
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2answers
2k views

A new kind of fractal?

http://www.gibney.de/does_anybody_know_this_fractal Is this some known kind of fractal? Update: This one got a lot of great feedback from around the net. I summarized it here: ...
1
vote
3answers
153 views

Convergence of the arguments of a sequence of complex numbers

Suppose the sequence $z_{n}$ converges to a nonzero limit $A$ and let $\Phi_{n}$ be any sequence of values of $Arg (z_{n})$ satisfying the inequality $$|\Phi_{m}-\Phi_{n}|<\pi$$ for $m>N, ...
1
vote
0answers
55 views

Proving that two representations of a Fourier series are the same

I have to show that $$\sum_{n=0}^\infty A_n\cos\left({xn\frac{2\pi}{T}-\theta_n}\right) \equiv \sum_{n=-\infty}^\infty c_n \mathrm{e}^{\left({ixn\frac{2\pi}{T}}\right)}$$ I have tried two ...
1
vote
1answer
51 views

Polynomial $P\in\mathbb{R}[x]$ with $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$

I have already shown that any polynomial $P\in\mathbb{R}[x]$ satisfies $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$ My question is, given a polynnomial $P\in\mathbb{C}[x]$, how I can ...
0
votes
1answer
124 views

Convergence of the arguments of a convergent sequence

Prove that if the sequence $z_{n}$ converges to a nonzero limit $A$ which is not a negative real number, then $\arg z_{n}\to \arg A$, where finite number of terms of $z_{n}$ which may wanish are ...
0
votes
2answers
95 views

solve complex conjugation equation

Find real part of $\left(\frac{a+bi}{a-bi}\right)^2-\left(\frac{a-bi}{a+bi}\right)^2$, for any $a,b$ (they are real numbers) Please help guys
5
votes
2answers
433 views

Proof of an inequality about $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$

I've encountered an inequality pertaining to the following expression: $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$, where $z$ is a complex number. After writing $z$ as $x + iy$ we have ...
4
votes
3answers
168 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
1
vote
2answers
101 views

Proving an inequality concerning arbitary complex numbers

$\def\abs#1{\left|#1\right|}$If $y$ and $z$ are any complex numbers then prove that \[ 2 \abs{y+z}\ge \bigl(\abs y + \abs z\bigr) \abs{\frac y{\abs y} + \frac z{\abs z}} \]
1
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4answers
88 views

Complex functions

I need help solving this problem: For $z$ a complex number, let $g(z) = \frac{1 + 2z }{1 + z}$. Find a function $h_1(z)$ such that $h_1(g(z)) = z$ and another (possibly the same) $h_2(z)$ such that ...
1
vote
2answers
166 views

Cauchy's formula: $\int_\gamma \frac{e^{z}}{z(z-3)} dz $

trying to compute the integral $\int_\gamma \frac{e^{z}}{z(z-3)} dz $, where $\gamma:[0,2\pi]\to\mathbb{C}, \gamma(\theta)=2e^{i\theta} $ but not sure where to begin. I know, from Cauchy's formula, ...
1
vote
2answers
67 views

If $N(\alpha)$ = $p$ where $p$ is an odd prime, then is ($\alpha, \bar\alpha$) = $\mathbb{G}$?

Here, $N(\alpha)$ stands for the norm of $\alpha\in\mathbb{G}$, $\mathbb{G}$ is the set of Gaussian Integers, and ($\alpha, \bar\alpha$) is the ideal generated by $\alpha$ and $\bar\alpha$. In other ...
2
votes
2answers
121 views

How can I break up $z = \frac{5}{9+3i}$

Into its real and imaginary components? Wolfram tells me it's equivalent to $\frac{1}{2}+\frac{i}{6}$, but I don't know how to arrive there myself. Thank you!
1
vote
1answer
124 views

complex numbers and euler's rule

I have an equation in the time domain $$A||H(s)||\sin(\omega t+\angle H(s))$$ I understand I can use the Euler equation for the sin term here. However the solution I have says the result after ...
1
vote
1answer
382 views

Solve the inequality $|z^2|-|z|\ \Re(z)>0$

Determine the set of complex numbers $z$ such that $|z|^2-|z|\ \Re(z)>0$ This is my process: Putting $z=x+iy$, we have $\Re(z)=x$ (real part), $|z^2|=x^2+y^2$, $|z|=\sqrt{x^2+y^2}$, and: ...
3
votes
4answers
635 views

Calculate a sum involving nth root of unity

Calculate $$1+2\epsilon+3\epsilon^{2}+\cdots+n\epsilon^{n-1}$$ Where $\epsilon$ is nth root of unity. There is a hint that says: multiply by $(1-\epsilon)$ Doing this multiplication I get: ...
0
votes
1answer
180 views

Complex n-th root question

Let $m$ and $n\neq0$ be any two integers.Show that $z^{m/n}=\left(z^{1/n}\right)^m$ has $n/(n,m)$ distinct values, where $(n,m)$ is the greatest common divisor of $n$ and $m$. Prove that the sets of ...
0
votes
1answer
263 views

Existence of limit of a function of complex variable

Let $a_0, a_1,\ldots,a_{n-1}, a_n$ complex numbers with $a_n\neq 0$. If $$f(z)=\left|a_n+\frac{a_{n-1}}{z}+\cdots+\frac{a_{0}}{z^{n}}\right|$$ Exist $\,\,\,\,\displaystyle{\lim_{|z|\rightarrow ...
0
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1answer
133 views

Problem of modulus of complex numbers

If $a$ is a complex number $a\neq 0$. Under that conditions $|a|\leq |a+z|$ with $z$ complex number?
0
votes
3answers
155 views

Inequality of complex numbers modulus

Someone could help to prove the following inequality of modulus of complex numbers: If $a\in\mathbb{C}$ then $$|a|\leq|a+z| \qquad \forall z\in\mathbb{C}$$
2
votes
1answer
133 views

Prove identity involving powers and trigonometric functions

Need help proving that: $$(1+\cos\alpha+i\sin\alpha)^{n}= 2^{n}\cos^{n}\frac{\alpha}{2}\left(\cos\frac{n\alpha}{2}+i\sin\frac{n\alpha}{2}\right)$$
1
vote
2answers
570 views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
1
vote
4answers
114 views

Need help to simplify the expression involving powers

$$\left(1-\frac{\sqrt{3}-i}{2}\right)^{24}$$ somehow this should be equal to :$$\left(2-\sqrt{3}\right)^{12}$$ but I can't see how...
2
votes
1answer
104 views

Complex logarithm and injectivity

Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex ...
2
votes
1answer
986 views

Mapping circles using Möbius transformations.

I need some help with the following problem from Ahlfors' Complex Analysis. Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where ...
0
votes
1answer
63 views

Definition of $\operatorname{arg}(z)$ choosing value of $\operatorname{Arg}(z)$

On what curve is $\arg (z)$ discontinuous if it is defined as the value of $Arg(z)$ satisfing the inequality: $$|z|-2\pi<\operatorname{Arg}(z)\leq|z|$$ would it be a ray from the origin with ...
1
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0answers
34 views

I need to describe a behavior of a curve depending one one parameter in z-plane

$|z^{2}-1|=\lambda$ My aprouch: taking $z=x+iy$ $$z^{2}=(x^{2}-y^{2})+i(2xy)$$ Then : $$|z^{2}-1|^{2}=x^{4}-2x^{2}y^{2}+2y^{2}-2x^{2}+y^{4}+1+4x^{2}y^{2}=\lambda^{2}$$ Now taking ...
11
votes
3answers
530 views

Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
2
votes
1answer
57 views

Familys of curves in z-plane depending on 1 parameter

Describe the family of curves depending on $C>0$ $$\left|\frac{z-z_{1}}{z-z_{2}}\right| = C $$ and $$arg\frac{z-z_{1}}{z-z_{2}} = C $$ What I got: let $z=x+iy, z_{1}=a+ib, z_{2}=c+id$ ...
49
votes
3answers
2k views

Why is $i! = 0.498015668 - 0.154949828i$?

While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!. Curiously, Google's ...
0
votes
1answer
185 views

For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?

For example, when $\alpha = 2$, $\ln(z^{2}) \neq 2\ln(z)$, because argument z is determined up to constant $2 \pi k$. So $$ \ln(z^{2}) = \ln(z) + \ln(z) = \ln(z_{k_{1}}) + \ln(z_{k_{2}}) \neq ...
0
votes
3answers
195 views

Describe the locus of points of this fraction

What's the locus of: $$\mathrm{Im} \bigg(\frac{z-z_{1}}{z-z_{2}}\bigg)=0$$ I tried to figure it out and I get that it's a line. But it looks to me it will be wrong.
1
vote
3answers
173 views

Solving roots of complex number

Given $z= \dfrac{i-1}{2}$. Evaluate $z^{1/2}$ and show the roots. I got $z^{1/2}=\dfrac{1}{2^{1/4}}\left(\cos\dfrac{3\pi}{8} + i\sin\dfrac{3\pi}{8}\right)$ First of all is my $z^{1/2}$ correct? And ...
2
votes
2answers
83 views

Is every complex number limit of a sequence of the form $\sin z_n$

What are the values of $z\in\mathbb{C}$, such that there is a non-constant sequence $z_n\in\mathbb{C}$ and $\sin z_n\to z$ ? How to find such a sequence if it exists ?
7
votes
5answers
252 views

I don't understand $\sqrt{-9i}$.

I try to visualise it on a graph, where x is real numbers and y is the imaginary numbers. $\sqrt{9} = (3,0)$ and $(-3,0)$. $\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = (0,3) $ and $(0,-3)$. ...
1
vote
1answer
51 views

Calculation, disc in $\mathbb{C}$ [closed]

I have a little (annoying) problem. Suppose we have a disc $B_{r}(a)$ in $\mathbb{C}$. Let the center be $a$ and radius $r$. Then you have a point $t$ on the boundary $\partial B_{r}(a)$ and ...
3
votes
2answers
573 views

How to go about solving $((1+iz)/(1-iz))^4 = 1/2 + i\sqrt3/2$?

I have problem solving this equation: $$ \left(\frac{1+iz}{1-iz}\right)^4 = \frac12 + i {\sqrt{3}\over 2} $$ I know how to solve equations that are like: $$ w^4 = \frac12 + i {\sqrt{3}\over 2} $$ ...
2
votes
4answers
867 views

Vertices of equilateral triangle inscribed in the unit circle

Prove that if $z_{1}+z_{2}+z_{3}=0$ and $|z_{1}|=|z_{2}|=|z_{3}|=1$ then the points $z_{1},z_{2},z_{3}$ are the vertices of an equilateral triangle inscribed in the unit circle $|z|=1$. My idea was ...
13
votes
5answers
598 views

Can I keep adding more dimensions to complex numbers?

I know about the concept of the complex plane, and I was amazed to find out that you're basically rotating numbers around this plane by multiplying by i, but, is there a way to jam the third dimension ...
0
votes
4answers
226 views

Prove a relation of arguments of 3 complex numbers with equal modulus

Prove that $$\arg \left(\frac{z_{3}-z_{2}}{z_{3}-z_{1}}\right) = \frac{1}{2} \arg\left(\frac{z_{2}}{z_{1}}\right)$$ if $|z_{1}|=|z_{2}|=|z_{3}|$.
3
votes
3answers
176 views

How to calculate $(3+4i)\cdot(1+i)$

I have recently read an article on imaginary numbers. It was very interesting, but left me with the above question. It had the answer in the question, it was $-1+7i$. But how do I calculate this?
25
votes
5answers
1k views

Is the square root of -1 rational?

This is not a deep question, but if there is a definite answer then here is the place where I will find it. Is it justified to say that $i =\sqrt{-1}$ is rational? The origin of this question lies ...
2
votes
2answers
71 views

Sorting $i$ amongst integers

In the game Wits & Wagers, players each answer a numerical question and the answers are then sorted in ascending order for scoring, details of which being irrelevant for this question. In this ...
1
vote
1answer
376 views

Complex Numbers in Fractal Algorithms

I am a high school freshman who is undertaking a small development project on fractals. I do not want to get too in depth, but I would love to blow my math teacher's socks off. Having looked through ...
300
votes
18answers
55k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
3
votes
1answer
109 views

a complex analysis problem

Let $\alpha ,\beta$ be two complex numbers with $\beta \neq 0$ , and $f(z)$ a polynomial function on $\mathbb{C} $ such that $f(z)=\alpha$ whenever $z^5 = \beta$. What can you say about the degree ...
4
votes
4answers
194 views

$(1+i)$ to the power $n$ [duplicate]

Possible Duplicate: Complex number: calculate $(1 + i)^n$. I came across a difficult problem which I would like to ask you about: Compute $ (1+i)^n $ for $ n \in \mathbb{Z}$ My ideas so ...
0
votes
3answers
81 views

Circle in a complex plane.

Let $C$ be a circle in the complex plane, and let $x$ be a fixed, non-zero complex number. Prove that $\{xz : z \in C\}$ is also a circle. I would really appreciate any help that would get me ...
1
vote
3answers
400 views

complex number question involving modulus

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\frac{|z − w|}{|1 − z^*w|} = 1$$ [Hint: Note that $|a|^2 = aa^*$.] Hey guys, couldn't get my ...
1
vote
2answers
162 views

Row reduction over any field?

EDIT: as stated in the first answer, my initial question was confused. Let me restate the question (I have to admit that it is now quite a different one): Let's say we have a matrix $A$ with entries ...