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

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Complex Number inqualities

Although the inequalities are not defined on complex numbers. But does the inequality $x < 4 + 5i$ be said to possess any solutions ? Where $ i = \sqrt{-1}$.
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40 views

Can some one explain why the answer to part a describes a circle, or part of it?

Problem Statement: The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by $w=\frac{z+1}{z+i}, z\neq i$. a) Show that $T$ maps points on the half-line $\arg ...
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2answers
28 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
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1answer
28 views

Why is (-1)^(2/3) equal to -1/2+(i sqrt(3))/2

Can someone please explain to me how $(-1)^{\frac2 3}$ can be written as $\frac {-1}{2}+\frac{i \sqrt3} 2$ ? Do you use the corrolation $(-1)^c = e^{(i c \pi)}$, where ${c}$ is a constant?
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4answers
29 views

Find Solution of trigonometric complex equation

Find the solutions of $\sin z = 3$ There are 2 ways to solve this, I know how to do this with: $\sin z = \frac{1}{2i}(e^{iz}-e^{-iz}) = 3$ Now, I am now doing in the way: $\sin z = \sin x \cosh y+i ...
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3answers
64 views

Is $(a+bi)(a-bi) = a^2 + b^2 $ solely a real number or a complex number?

I have not dealt with complex numbers for a while now, but I was wondering if I multiplied the complex number $a+bi$ by its conjugate $a-bi$ to obtain $$(a+bi)(a-bi) = a^2 + b^2 $$ where $a,b \in ...
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1answer
31 views

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients?(complex-number)

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients? Its coefficient is $1$ and $1$ is a real number, isn't it?
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2answers
23 views

Complex Polynomials

I have this question on a practice final, Find all the solutions to the equation $$2z^2 = √2 − i√2$$ I'm not quite sure how to solve this! Should I approach this with synthetic division? Any help ...
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3answers
41 views

What is a basis for the vector space $ \Bbb{C}^{n} $ (a complex vector space)?

I know that a basis for $ \Bbb{C} $ is $ \{ 1,i \} $. This set is linearly independent in $ \Bbb{C} $ and spans $ \Bbb{C} $. I think that the dimension of $ \Bbb{C}^{n} $ may be $ 2 n $, but I’m just ...
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0answers
13 views

equvivalence resistance of hexagonal infinite

I am trying to evaluate equivalence resistance between two nodes of hexagonal infinite grid, I am stuck at the integral at end of the image attached. pl see if the integral could be evaluated. Let ...
2
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1answer
29 views

Complex integration on circle

Calculate the integral of $g(z)$ along the closed path $|z-i|=2$ in the positive direction when i)$g(z)=\frac{1}{z^2+4}$ ii)$g(z)=\frac{1}{(z^2+4)^2}$ First I checked the described area ...
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5answers
291 views

Confused with imaginary numbers [duplicate]

In 9th grade I had an argument with my teacher that ${i}^{3}=i$ where $i=\sqrt{-1}$ But my teacher insisted (as is the accepted case) that: ${i}^{3}=-i$ My Solution: ${i}^3=(\sqrt{-1})^3$ ...
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1answer
24 views

Complex integration and theorems

If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$ show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out ...
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2answers
22 views

Integral of strictly real function has imaginary component

Intuitively and informally speaking, $\int_{a}^{b}f(x)dx$ is summing all of the values $f(x)$ yields for $x\in [a,b]$. So it would make sense that if $f(x)$ is strictly real over $[a,b]$, then ...
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1answer
25 views

Complex function, analyticity domain

Find the function domain of analyticity i)$f(z)=\frac{z^2}{z-3}$ ii)$f(z)=ze^{-z}$ To check the domain of analyticity of a function, I only need to replace $z=x+iy$ and check the conditions of ...
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2answers
28 views

Simple math Question concerning the natural logarithm of Complex Number

There is this simple exercise, in which the complex number is given in polar form as z= mod=|10|,arg=322.75 degrees and i must find the ln of it. So to do that i must first convert the complex number ...
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3answers
100 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 ...
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1answer
16 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, ...
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1answer
38 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} ...
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1answer
38 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 ...
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1answer
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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 = ...
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3answers
62 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 ...
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1answer
17 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 ...
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0answers
18 views
+50

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 ...
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2answers
48 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 ...
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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 ...
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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" ...
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1answer
15 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 ...
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1answer
25 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
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1answer
21 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 ...
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3answers
51 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) ...
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3answers
46 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.
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3answers
52 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?
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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: ...
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2answers
44 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) ...
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0answers
23 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 ...
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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 ...
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2answers
37 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 ...
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1answer
30 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 ...
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2answers
23 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 ...
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4answers
28 views

finding roots of cubic equation and the values of constants [closed]

$x^3+px^2+qx+30=0$ where $p$ and $q$ $\in R$, has a root $1+2i$. $1)$ Find the other non-real root. $2)$ Find the third root of the equation. Hence, or otherwise, find the values of $p$ and $q$.
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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 ...
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1answer
36 views

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

How can I solve for the complex solutions of $$ x^3 = -1 $$
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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
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1answer
26 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, ...
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1answer
52 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
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0answers
33 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
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2answers
19 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?
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2answers
40 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
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2answers
59 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 ...