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

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17
votes
2answers
1k views

De Moivre's Theorem. Motivation and origins.

I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous $$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$ but ...
1
vote
4answers
287 views

Finding the n-th root of a complex number

I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$. So What I have so far is : $$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$ $$r = 1,\ \theta = \frac{\pi ...
1
vote
1answer
32 views

Trying to figure out a complex equality

An answer to a comlex equation I was working on was $$z = \frac{1}{2} + \frac{i}{2}$$ My teacher further developed it to be $$e^{\frac{i\pi}{4}-\frac{1}{2}\ln{2}}$$ And here's what I tried: $$z = ...
1
vote
2answers
97 views

simplify complex polynomial $p(t) \in \mathbb{C}[t]$

How to simplify the following polynomial? $$ \begin{align} (t - \sqrt{3} \; e^{ \frac{\pi}{3} i }) (t - \sqrt{3} \; e^{ -\frac{\pi}{3} i }) &= t^2 - \sqrt{3} \; e^{ \frac{\pi}{3} i } \; t - ...
3
votes
3answers
176 views

$(\cos \alpha, \sin \alpha)$ - possible value pairs

We introduced the complex numbers as elements of $ \mathrm{Mat}(2\times 2, \mathbb{R})$ with $$ \mathbb{C} \ni x = \left(\begin{array}{cc} a & -b \\ b & a \\ \end{array}\right) = ...
4
votes
3answers
168 views

Algebraic complex numbers $z$ satisfying the equations $z^n+\bar{z}^n = z+\bar{z}$ for all positive $n$

Is there a complex number $z$ such that $z\neq 0,1$ and $z^n +\bar{z}^n = z+\bar{z}$ for all positive integers $n$? Is there an algebraic complex number $z$ such that the above properties hold? The ...
-4
votes
1answer
150 views

What is the conjugate of $\frac{1}{2}+ \frac{3}{2}i$?

What is the conjugate of $\dfrac{1}{2}+ \dfrac{3}{2}i$? Firstly, what is conjugation? And secondly, can you should the steps to doing this? "$i$" is the imaginary unit.
1
vote
2answers
163 views

How do you show $x_n=n(e^{\frac{2\pi i}{n}}-1)$ converges or not in C with the usual norm?

I have taken the limit of $x_n$ and got $2i\pi$. Now I am stuck trying to show $|n(e^{\frac{2\pi i}{n}}-1)-2i\pi|=0$. I am thinking I should try to write this in the form of $|a+bi|$ but I can't ...
1
vote
1answer
153 views

Von Mises width at half height

I'm fitting the following Von Mises type function to some data: $(A/2\pi)e^{k\cos(\theta)}+C$ where A and k are positive. I want to calculate the width at half height from the lowest point of the ...
1
vote
1answer
105 views

What are the values of $(-1)^{2i}$? [duplicate]

Possible Duplicate: What is the value of $1^i$? I read that $$ (-1)^{2i}=\exp(2i\log -1)=\exp(-2\pi-4\pi k) $$ for $k\in\mathbb{Z}$. How does the second equality follow? I calculate ...
4
votes
1answer
410 views

Using the complex logarithm to find the sum of angles in a triangle.

Suppose you have a triangle with vertices $a$, $b$, and $c$. I asked earlier how you can define the angles in a triangle based on the $\log$ function. I received the answer that, for instance, the ...
2
votes
2answers
205 views

How to find complex numbers $z,\lambda,\mu$ such that $(z^\lambda)^\mu\neq z^{\lambda\mu}$

Let $z$, $\lambda$, $\mu$ be complex numbers. Find a case where $(z^\lambda)^\mu$ is not equal to $z^{\lambda\mu}$. In our book, $a^b = \exp( b \cdot \operatorname{Log}(a) )$. ...
4
votes
1answer
1k views

Proving the Schwarz Inequality for Complex Numbers using Induction

I want to prove the following version of the Schwarz Inequality for complex numbers $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $b_1, b_2, \ldots, b_n \in \mathbb{C}$: $$|\sum_{j=1}^n a_j ...
2
votes
1answer
371 views

Image of a map in the complex plane

Is there an elegant way (either intuitive/ by a series of diagrams or by manipulating numbers/algebra) to find out what the image of $\sin(w)$ where $w\in \mathbb C$ from a domain say $\{w\in \mathbb ...
2
votes
2answers
371 views

Undiscovered fractal sets?

I'm interested in the topic of fractals, such as those created by the borders of the Mandelbrot and Julia sets. My question is if there are other, not yet discovered fractal sets, which one could ...
2
votes
1answer
59 views

A problem with polynomials.

This is a problem from a test in my course in analytic functions. I didn't manage to solve it. Could you please give me a hint? The problem is: Calculate the third root of the sum of the coefficients ...
5
votes
4answers
553 views

Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
1
vote
3answers
736 views

Simple expressions for $\sum_{k=0}^n\cos(k\theta)$ and $\sum_{k=1}^n\sin(k\theta)$? [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in A.P? I'm curious if there is a simple expression for $$ 1+\cos\theta+\cos 2\theta+\cdots+\cos n\theta ...
7
votes
3answers
670 views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
2
votes
4answers
675 views

finding roots of complex equation

I have here a complex equation: $$z^2 - (7+j)z + 24 +j7 = 0$$ How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much ...
4
votes
1answer
555 views

Why is the argument of $i$ equal to $\pi/2$?

So it's obvious geometrically that the argument of $z=i$ is $\pi/2$. However the method of getting the argument is $\arctan(y/x)$. And when in the case of $z=i$, $y/x = 1/0$ which is undefined... So ...
0
votes
1answer
118 views

terminology: euler form and trigonometric form

Am I right, that the following is the so-called trigonometric form of the complex number $c \in \mathbb{C}$? $|c| \cdot (\cos \alpha + \mathbf{i} \sin \alpha)$ And the following is the Euler form of ...
1
vote
3answers
96 views

Evaluate a complex set

Can you please help me finding an exact description of the set: $$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$ For any $0<R \in \mathbb{R}$. My feeling is the $E_R = \mathbb{C}$, for any ...
1
vote
3answers
2k views

primitive n-th roots of unity

Show that the primitive n-th roots of unity have the form $e^{2ki\pi/n}$ for $k,n$ coprime for $0\leq k\leq n$. Since all primitive n-th roots of unity are n-th roots of unity by definition they all ...
0
votes
1answer
94 views

rearrange $z \mapsto z^2 + c$

Mathematics, some of its magic is that a lot is known about how to rearrange its statements (equations). Given the Mandelbrot Set: $z \mapsto z² + c$ (or more precisely) $z_{i+1} = z_i ^2 + c$ ...
2
votes
2answers
130 views

That $|a|\leq|b|$ implies existence of complex $z$ satisfying $|z-a|+|z+a|=2|b|$?

I'm looking at the equation $|z-a|+|z+a|=2|b|$. If there are complex values $z$ satisfying this equation, then $$ 2|b|=|z-a|+|z+a|=|a-z|+|z+a|\geq|(a-z)+(z+a)|=|2a|=2|a| $$ so $|a|\leq |b|$. ...
1
vote
2answers
329 views

Division of Complex Numbers

Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show ...
1
vote
1answer
550 views

Inductive proof of Cauchy's inequality for complex numbers?

I'm trying to put together an inductive proof of Cauchy's inequality for the complex case, $$ \left|\sum_{i=1}^na_ib_i\right|^2\leq\sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2. $$ The base case is easy, ...
10
votes
7answers
2k views

Is there a formula for $(1+i)^n+(1-i)^n$?

I'm wondering if there is a formula for the value of $(1+i)^n+(1-i)^n$? I calculated the first terms starting with $n=1$ to be, in order, $2$, $0$, $-4$, $-8$, $-8$, $0$, $16$, $\dots$ So it seems ...
0
votes
1answer
339 views

If $\operatorname{Re}^{2}(x)=-1$, what is $x$?

$i=\sqrt{-1}$ $\operatorname{Re}(z)+i\cdot\operatorname{Im}(z)=z$ If $\operatorname{Re}^{2}(x)=-1$, what is $x$? $x$ cannot be defined in complex number as $(a+ib)$. { $a$ and $b$ are real numbers ...
2
votes
1answer
50 views

Stability of a set under multiplication and conjugaison / Inversible elements of a set

Let $j=e^{2i\pi/3}$. How can I prove that the set $A=\mathbb{Z}+j\mathbb{Z}$ is stable under multiplication and conjugaison ? If I understood the question well, I need to prove the following : a)If ...
2
votes
1answer
404 views

Bound the complex roots of a polynomial above

We consider $P(z)=a_{0}+a_{1}z+\cdot+a_{n-1}z^{n-1}+a_{n}z^n$, with $a_{0},\ldots,a_{n-1},a_{n} \in \mathbb{C}$ and $a_{n}\neq0$. Let $R=\max_{0\leq k\leq n-1}\left | \frac{a_k}{a_n} \right |$ and ...
0
votes
0answers
91 views

Uniqueness of homography

Let $(z_{1},z_{2},z_{3})$ and $(z'_{1},z'_{2},z'_{3})$ be two $3$-tuples of complex coordinates of non collinear points. How can I prove that there exists a unique homography $h$ such that ...
2
votes
1answer
254 views

Inequality for modulus

Let $a$ and $b$ be complex numbers with modulus $< 1$. How can I prove that $\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ? Thank you
0
votes
1answer
32 views

NSC for boundedness of a complex sequence

I suggest this problem to you. Let $\theta \in \mathbb{R}$. For all nonnegative integer $n$ let $s_{n}=1+\exp(i\theta)+\exp(2i\theta)+\ldots + \exp(in\theta)$. Determine a necessary and sufficient ...
1
vote
1answer
626 views

Relation with sum of modulus of complex numbers

Let $z_{1}, z_{2}, ..., z_{n}$ be nonzero complex numbers, with $z_{k}=p_{k}\exp(i\theta_{k})$, where $p_{k}$ is a positive real number and $\theta_{k}$ real. Can you help me prove that $\left | ...
4
votes
1answer
105 views

Existence problem for a polynomial with complex coefficients

Let $n$ be a nonnegative integer and $a_{0}, a_{1}, ..., a_{n}$ real numbers. For any real number $t$ let $f(t)= \sum_{k=0}^{n}a_{k}\cos(kt)$. Could you help me with the following two questions ? a) ...
0
votes
2answers
520 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
2
votes
1answer
142 views

Visualizing the flat complex conic

Consider a conic $X^2 + Y^2 - Z^2 = 0$ in $\mathbb{C}P^2$. In an affine chart $Z \neq 0$ it is supposed to look like a circle (however it looks in $\mathbb{C}^2$), but the deceptiveness of imagining ...
1
vote
2answers
315 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
4
votes
1answer
114 views

Geometric/Simpler proof for the following complex numbers problem

I wonder if there is a geometric proof or a short proof of the following: let $z_1,z_2,z_3$ be three complex numbers of modulus $r$. prove that the number $$ ...
6
votes
1answer
307 views

Why are primitive roots of unity the only solution to these equations?

I was led by this question to the following problem: Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy $$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 ...
13
votes
4answers
3k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
0
votes
2answers
2k views

Complex conjugate of function

I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants. I am trying to find the probability density, so I need to find the product of $\psi$ ...
13
votes
6answers
787 views

How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?

Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$? Calculating the complex roots and then the result could be an answer I suppose, but ...
1
vote
4answers
271 views

solve complex equation

$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...
5
votes
1answer
973 views

Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$

The question I am working on starts of with: Find the five fifth roots of unity and hence solve the following problems I have done that and solved several questions using this, however ...
4
votes
1answer
167 views

Complex number inequality?

Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that $$ \frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}? $$
2
votes
2answers
110 views

How is $\mathbb{H}$ a $\mathbb{R}$-Algebra, but not a $\mathbb{C}$-algebra?

It says that $\mathbb{C}$ is not in the center of $\mathbb{H}$. Definition of $\mathbb{K}$-algebra for a ring if $Z(R)=K$. However, you can do this $(a+bI+cJ+dK)(e+fi)=(e+fi)(a+bI+cJ+dK)$. So I don't ...
6
votes
3answers
369 views

Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute

As the question indicates we are supposed to find the modulus of z-1. When trying to solve the problem I drew a diagram which you can see below: The book I am working in solved a similar problem ...