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

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Simplifying an function with complex numbers

Given that $h_i, h_d $ are $\in \mathbb{C}$ Can the following be further simplified $$h_dh_d,+h_ih_i+h_dh_d^*+h_ih_i^*\leq 2h_ih_d+h_ih_d^*+h_dh_i^*\tag 1$$ I know that (1) can be simplified into ...
10
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11answers
197 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$?

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
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0answers
8 views

How to calculate the variance of the argument of a complex number?

Given a number $s \in \mathbb{C}$ and the (Gaussian) variances of its components $\sigma^2(\Re(s))$ and $\sigma^2(\Im(s))$, how can I calculate the variance $\sigma^2(\arg(s))$ and the covariances ...
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4answers
119 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
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1answer
26 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
11
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3answers
172 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
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2answers
40 views

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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2answers
41 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
29 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
32 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 ...
2
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3answers
65 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 ...
2
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1answer
32 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
30 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 ...
5
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5answers
295 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
102 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
17 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
33 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
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3answers
63 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
20 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 ...
3
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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
22 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
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.
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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: ...
0
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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 ...
-1
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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 $q$.
2
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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
37 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} ...