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

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5
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3answers
109 views

Does $1^i$ and $1^{\frac{0}{0}}$ also give $1$ again? [duplicate]

This is the property of Real number $1$ that, $1^n=1$ does this property only hold $\forall n \in \mathbb R$ or also $1^i=1$ and $1^{\frac{0}{0}}=1$ If it is; explain how? I think that it should ...
1
vote
1answer
19 views

$(N(\alpha), N(\beta)) = 1 \rightarrow (\alpha, \beta) = 1$ and backwards?

Let us have $\alpha, \beta$ arbitrary Gaussian integers. Is it true, that if $(N(\alpha), N(\beta)) = 1 \rightarrow (\alpha, \beta) = 1$? Is it true backwards? I know when a Gauss-integer is prime, ...
0
votes
1answer
40 views

Solve Trigonometric Complex Equation

Find all solutions of $\sin (z) = 2$. Here are the things I did: 1) By definition: $\sin z =\dfrac{e^{iz} − e^{−iz}}{2i}= 2$. Multiply $2i$ to the equation and make it quadratic: $e^{2iz} ...
0
votes
1answer
40 views

Sum of bits in range of twindragon curve

http://blog.garritys.org/2012/12/base-i-1-there-be-dragons.html As the link above shows, it's possible to represent every Gaussian integer by converting a number N into its binary representation and ...
-1
votes
1answer
37 views

Solve $3x³ + 3y³ + 2x² - 32 = 0$, $4x² + 2 = 0$ and $10y² + 2x² + 12 = 12x³$.

Hi my friend asked this to me, i'm not good at math. $$3x³ + 3y³ + 2x² - 32 = 0$$ $$4x² + 2 = 0$$ $$10y² + 2x² + 12 = 12x³$$ remove 2x² $$2x² = -1$$ $$3x³ + 3y³ - 1 - 32 = 0$$ $$10y² - 1 + 12 = ...
3
votes
3answers
70 views

Zeroes of sin(x)

Consider the function f = $\sin(x)$ defined as $$ \sin(x) = \frac{e^{ix}- e^{-ix}}{2i} $$ How to prove that the only zeroes of this function lie on the line $i = 0$ in the complex plane and ...
1
vote
1answer
18 views

There exists a continuously differentiable bijection, $g:[a,b]\to [c,d]$ satisfying $g'(k)>0$ with $z(k)=w(g(k))$

Let $z:[a,b]\to \mathbb{C}$ and $w:[c,d]\to \mathbb{C}$ such that there exists $t(s):[c,d]\to [a,b]$ which is a continuously differentiable bijection with $t'(s)>0$ and $w(s)=z(t(s))$. Then I ...
3
votes
0answers
99 views

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
0
votes
2answers
51 views

Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$

Let $[z_1,z_2,z_3,z_4]$ denote the cross ratio of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized ...
1
vote
2answers
53 views

Simplify $Im \left(\frac{az+b}{cz+d}\right)$

Let $z \in \mathbb{H}$, where $\mathbb{H}$ denotes the half plane $\mathbb{H}=\{z \in \mathbb{C}:Im(z)>0\}$. Let \begin{equation*} f(z)=\frac{az+b}{cz+d} \end{equation*} which is called a Mobius ...
3
votes
0answers
47 views

Why is there only one type of imaginary number? [duplicate]

We've defined the square root of -1 as an imaginary number i (or j, if you're a physicist). Is there any reason why we can't/haven't made other systems of imaginary numbers for other "impossible" ...
1
vote
1answer
16 views

Difference of roots of unity in polar form

I want to write the difference between $n$-th roots of unity in the form $re^{i \theta}.$ It is enough to find the polar form of $1 - \zeta^k$. By thinking geometrically, I can guess the formula $$1 ...
4
votes
1answer
28 views

Proving for $w \in \Bbb C$ with modulus $1$ and argument $2 \theta$ that $\frac{w-1}{w+1}=i\tan \theta$

The complex number w has modulus $1$ and argument 2$\theta$ radians. Show that $$\frac{w-1}{w+1}=i\tan \theta.$$ Attempted solution: I just assumed that $w=1(\cos 2\theta +i \sin 2\theta)$ and ...
1
vote
1answer
26 views

Exponent identities with imaginary exponents$\left(a^i\right)^i$

I've been trying to understand how imaginary exponents work, and I think I mostly understand it, but I'm confused by something like $\left(a^i\right)^i$ (where $a$ is real). According to the normal ...
2
votes
3answers
56 views

Finding the sum of the trigonometric serie:

There are two series: $$1) 1+\dfrac{\cos{x}}{p}+\dfrac{\cos{2x}}{p^2}+...+\dfrac{\cos{nx}}{p^n}=\sum_{k=0}^{n}{\dfrac{\cos{kx}}{p^k}}$$ $$2) ...
0
votes
3answers
47 views

Proving that $|z-1|-|z+1|=1$ its an hyperbola, and $\Re(1-z)=|z|$ its an ellipse.

Proving that $|z-1|-|z+1|=1$ its an hyperbola, and $\Re(1-z)=|z|$ its an ellipse. If $z\in \mathbb C$ I cant see why there are a hyperobla and an ellipse respct.
0
votes
3answers
54 views

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?, being $A,B \in \mathbb C$ How can it be described?
0
votes
3answers
44 views

Finding the minimum value of $|a+b\omega+c\omega^2|$ if $a,b,c$ are unequal integers where $\omega^3=1$

My try 1: $$|a+b\omega+c\omega^2|\le\sqrt{|a+b+c||\underbrace{1+\omega+\omega^2}_0|}$$ Cauchy-Scwartz won't give us an upper bound since $a,b,c$ are nonequal integers. My try 2: ...
0
votes
2answers
45 views

divisibility of complex numbers

I want to show that $(a + bi)|(c + di)$ is equivalent to the statement that the ordinary integers $(a^2 + b^2)|(ac + bd)$ and $(a^2 + b^2)|(-ad + bc)$. I also want to show that $(a + bi)|(c + di) ...
0
votes
0answers
24 views

continuity of the complex square root function

I want to show that there is no continuous square root function in the complex plane, i.e. a function $f:\mathbb{C}\rightarrow\mathbb{C}$ with $f(w)^2=w$ for all $w \in \mathbb{C}$. I already ...
12
votes
3answers
1k views

Finding the square root of a complex number - why two solutions instead of four?

I want to find the square roots of a complex number, $w = a+ib \in \mathbb{C}$, i.e. I'm looking for solutions, $z = x + iy$, for the equation $z^2 = w$. This question has been asked here a couple of ...
3
votes
2answers
38 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic integer $\beta$ Pisot-like if $|\beta|_{\mathbf{v}} > 1$ for the place $\mathbf{v}$ of $\Bbb{Q}(\beta)$ corresponding to the embedding $\beta ...
2
votes
1answer
32 views

For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

I used the ratio test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the ...
0
votes
2answers
28 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
2
votes
3answers
51 views

How can I solve the simultaneous equations that arise in solving $\cos(z)=2$.

If I have $\cos(z)=2$ I can say $\cos(a+ib)=2$ using double angle ideas $\cos(a)\cos(ib)+\sin(a)\sin(ib)=2$ using Euler's formula $\cos(a)\cosh(b)+i\sin(a)\sinh(b)=2$ equating real and imaginary ...
-3
votes
1answer
39 views

Solving $x^3 = -1$ for complex numbers [duplicate]

How can I solve for the complex solutions of $$ x^3 = -1 $$
0
votes
1answer
20 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
0
votes
1answer
28 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
0
votes
1answer
53 views

The value of $1+2\alpha+3\alpha^{2}+…+n\alpha^{n-1}$ for complex $\alpha$

Compute the value of $$1+2\alpha+3\alpha^{2}+...+n\alpha^{n-1}$$ in the form of a complex number where $\alpha$ is a non-real complex $n^{th}$ root of unity. The answer given is : ...
3
votes
0answers
35 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
0
votes
2answers
21 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
0
votes
2answers
41 views

What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument?

How is called the subset of Gaussian integers such that from all Gaussian integers having the same argument only one with the smallest absolute value is included? Is there a special name for them? ...
5
votes
2answers
60 views

Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.

Let , $f$ be entire function such that $|f\left(\frac{1}{n}\right)|\le \frac{1}{n^{3/2}}$ for all $n\in \mathbb N$. Then prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded. From the ...
0
votes
0answers
14 views

“Permutation” of squared norm and sum

In Problems and Solutions in Mathematics, 2nd edition, by Ta-Tsien, exercice 4312. Let $f$ be a periodic function on $\mathbb{R}$ with period $2 \pi$ such that $f|_{[0, 2 \pi]}$ belongs to $L^2(0, 2 ...
0
votes
1answer
59 views

Polynomial Equation Solution

Use Demoivre's theorem to show: $cos 7θ = 64 cos7 θ − 112 cos5 θ + 56 cos3 θ − 7 cos θ$ Hence,solve: $128x^7 −224x^5 +112x^3 −14x+1=0$ I've shown the first part and multiplied the equation by 2 and ...
-1
votes
1answer
40 views

What is the number of complex integers inside a circle of radius r? [closed]

What is the number of such complex integers, $z$, that $|z|\le r$? I am interested in a closed-form formula for integer $r$.
1
vote
3answers
60 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
1
vote
2answers
36 views

Expansion of imaginary numbers

If $(1+i)^{100}$ is expanded, what is the value of the real part of the result? I know that this has to do with binomial theory and Pascal's triangle, but I don't know how to use it here.
-2
votes
3answers
59 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
2
votes
0answers
27 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
1
vote
3answers
80 views

Simplifying a Complex Number

I have $\left ( \frac{e^{i\frac{\pi}{3}}}{1+i}\right )^{2014}$. I wish to simplify this to standard form. I simplify to $\left ( e^{i\frac{\pi}{12}} \right )^{2014}$ I can evaluate and simplify ...
-3
votes
2answers
74 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [closed]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
0
votes
2answers
45 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
1
vote
3answers
47 views

Is a matrix with complex entries invertable?

This is merely a question of interest and not for something I am doing in school. I have never seen a matrix with complex entries in class before, but mind you it was a limited linear algebra class, I ...
0
votes
2answers
38 views

Cauchy- riemann equations

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function that is differentiable at the point $z_0 =x_0 + iy_0$. Prove that $f'(z_0)= \frac{\partial u}{\partial x} (x_0,y_0) + i \frac{\partial ...
1
vote
0answers
39 views

How to use the for re^itheta to prove this?

Can someone please explain how to use the form $re^{i\theta}$ and de Moivre's to prove that: $$\sum_{n=1}^N \frac{\sin n\theta}{2^n} = \frac{2^{N+1} \sin \theta + \sin N\theta - 2\sin(N + ...
-1
votes
2answers
21 views

Representing a transformation from C to C with respect to the basis 1, i

I am having trouble understanding why the transformation: $ T(z) = (3+4i)z$ from C to C can be represented by the matrix $ \begin{bmatrix} 3, -4 \\ 4, 3 \end{bmatrix}$ with respect to the basis $ ...
0
votes
0answers
24 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
0
votes
1answer
31 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
1
vote
1answer
38 views

Prove that if $z$ is good then so is $z + r$ for every $r \in R$.

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ Note that $R$ is an integral ...