Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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7answers
473 views

Solve for $z$ in $z^5=32$

This was the last question on my Year 11 Complex Numbers/Matrices Exam Name all 5 possible values for $z$ in the equation $z^5=32$ I could only figure out $2$. How would I go about figuring this on ...
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1answer
82 views

Prove the following identities ..complex numbers

I found this question on my Algebra book but i couldn't answer it Can you please explain step by step Prove the following identities ..explain its geometric meaning $|1+z_1\bar z_2|^2 + |z_1-z_2|^2 = ...
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1answer
2k views

Equilateral triangle in complex plane [duplicate]

Prove that the points $a_1,a_2,a_3$ are vertices of an equilateral triangle if and only if $a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_3a_1$. I rewrite the equation as ...
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1answer
65 views

Calculate the following contour integral…

Calculate $$\oint_\gamma \frac 1{z-\sin z} dz$$ where the contour is the unit circle in the complex plane. I do not know how to find the order of the pole at 0, though I believe it is 3. Once I have ...
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1answer
83 views

Complex polynomial identity with norm condition

In this question, the following was shown: If $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in a complex variable $z$, satisfies the condition that $|R(z)|=1$ whenever $|z|=1$, then the ...
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1answer
197 views

I need to find the Image of the region

I need to find the Image of the region $\{z\in\mathbb{C}:\Re(z)>\Im(z)>0\}$ under the map $z\to e^{z^2}$ $z=x+iy\Rightarrow z^2=x^2-y^2+2ixy \Rightarrow ...
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1answer
41 views

form of groups of motions of tessellations

I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says: The triangle and hexagon tessellations have ...
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1answer
58 views

Complex number inequality

For any $z_1, z_2 \in \mathbb{C}$, is there exist $C>0$ such that $$ 4|z_1|^2 |z_2|^2 + |z_1^2 - z_2^2|^2 \ge C (|z_1|^2 + |z_2|^2)^2 \;\;?$$
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1answer
117 views

How to show that $f(z) = f(i)$

Exercise problem. I do not need a full solution because I am trying to solve myself. Just a hint would be great. Let $f$ be a polynomial with real coefficients. How to show that $f(z) = f(i)$ for ...
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1answer
161 views

Limits of complex error and gamma functions in the complex plane?

What are the following one-sided limits in the complex plane (in the form $x+iy$): For the complex error function: $\lim_{x \to 0^+, y \to 0^+}\text{erf}\left(x+iy\right) = $ $\lim_{x \to +\infty, ...
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1answer
126 views

Divergence of $\Gamma$ function for complex values

It is said that $\dfrac{1}{\Gamma(ix)}$ (of purely imaginary part) diverges. But why please?
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1answer
76 views

How can I practice Jean-Robert Argand idea of the rotation of a square root of -1

I am studying complex numbers and I really need an intuition on how they work. I found the following video of Mathematician named Adrien Douady https://www.youtube.com/watch?v=2kbM96Jr4nk He ...
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1answer
350 views

extended Euclidean (xgcd) in quadratic integer rings

Given a discriminant $D < 0$, I have the quadratic imaginary field $\mathbb{K} := \mathbb{Q}(\sqrt{D})$. And the quadratic integer ring is given by $\mathcal{O} = \mathbb{Z} + \mathbb{Z} \frac{D + ...
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1answer
88 views

What is the easiest way to define a complex number in exponential form in maple?

What is the easiest way to define a complex number in exponential form in maple? Is there a built-in function? eg: $\underline{Z} = 600 \cdot e^{-j45^\circ}$
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1answer
192 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
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1answer
39 views

similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.

Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane. What is the image of the curve : $y=e^x-x-1$ by the similarity $S$. My work : Let $z=x+iy$ ...
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1answer
57 views

Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]

Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows: $$ w = a+bi \\ z = c+di $$ Can we analogically write $$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
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2answers
534 views

How do you solve $z^4 = 2(1+i\sqrt{3})$

Solve $z^4 = 2(1+i\sqrt{3})$ in the form $r(\cos\alpha+i\sin\alpha)$ where $r>0$ and $0\le\alpha<2\pi$ I know you have to find $\arctan(\frac{\sqrt{3}}{1})=\frac{\pi}{3}$ and that is $\alpha$? ...
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1answer
118 views

What is the modulus of a number?

What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
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4answers
83 views

What is the polar form of $ z = 1- \sin (\alpha) + i \cos (\alpha) $?

How do I change $ z = 1- \sin (\alpha) + i \cos (\alpha) $ to polar? I got $r = (2(1-\sin(\alpha))^{\frac{1}{2}} $. I have problems with the exponential part. What should I do now?
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1answer
392 views

Form of periodic function involving exponential

I am trying to prove that if the function $f(z)= a_{1}e^{\lambda_{1}z} + ... + a_{n}e^{\lambda_{n}z}$ is periodic of period $T \neq 0$ with $a_{i} \neq 0$ for every $i$, then $\lambda_{i} = 2k_{i}\pi ...
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1answer
94 views

Lower bound for polynomial with complex coefficient

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$. ...
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1answer
106 views

Rotation on the unit circle K

If $\{a^{n}:n\in\mathbb{Z}\}$ is dense on the unite circle K, then $\{a^{n}:n< 0\}$ is also dense on K. How to prove this result?
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1answer
218 views

Find a complex number that satisfies the equation

Find one complex value of $x$ that satisfies the equation $\sqrt{3}\cdot x^7+x^4 +2=0$
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1answer
380 views

Continuous function on simple closed contour

Let $f$ denote a function that is continuous on a simple closed contour $C$. Using the Cauchy Integral formula, prove that the function $g(z)=\frac{1}{2\pi i}$ $\int_C$ $\frac{f(s)ds}{s-z}$ is ...
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1answer
131 views

$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform

I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple: $$ f(x) = \sin(x) $$ By definition its Fourier transform should be: $$ F(k) = ...
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2answers
43 views

Help please on complex polynomials

I wanted to know if there's any good approaches to these questions a)By considering $z^9-1$ as a difference of two cubes, write $1+z+z^2+z^3+z^4+z^5+z^6+z^7+z^8$ as a product of two real factors one ...
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1answer
109 views

Complex numbers and proof

I'm having trouble learning and understanding questions relating to complex numbers and was wondering if I can get any help. Thanks in advance for any help I can get! Given $x$ is a complex number ...
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1answer
245 views

Complex conjugate of a variable raised to the power $n$

What would be the complex conjugate for these three. Assuming $i$ is always $${\sqrt{-1}}$$ $$i^{11}$$ $$(2-3i)^3$$ $$\frac{3-i}{2i+5}$$
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1answer
38 views

Complex Variables…contour integral

For what values of $m$ and $n$ does $\int_C z^mz^{-n}dz=0$ and for what values does $\int_C z^mz^{-n}dz=2i\pi?$ I am stuck on this problem, any hint? Thanks
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1answer
143 views

What are the Complex (Non-Real) Eigenvectors of $3\times 3$ Rotation Matrices?

A $3\times 3$ rotation matrix $R$ that rotates $\mathbb{R}^3$ around the unit vector $v\in\mathbb{R}^3$ by angle $\theta$ (as defined by Rodrigues' rotation formula) satisfies the ...
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2answers
54 views

maths problem with absolute numbers and complex conjugates

With the following expression with complex conjugates $$ -xx^{*}+yy^{*} = -|x|^2 + |y|^2$$ can it be represented as $$|x|^2 - |y|^2$$ or since $ |x|^2 + |y|^2 = 2$ is it true that $|x|^2 - |y|^2 = ...
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1answer
378 views

How do I take the complex convolution of this impulse response and input?

I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of ...
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1answer
43 views

Proving $\arg(a)\equiv \alpha\Longleftrightarrow z_1z_2 \in \mathbb{R}$

Suppose the complex equation $iz^2+(2-i)az-(1+i)a^2=0$ as $a\in \mathbb{C}^{*}$. $z_1$ and $z_2$ are the solution of this equation and we have also $z_1*z_2 = a^2(i-1)$. How can I prove that ...
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1answer
272 views

Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials?

Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials ? Or does it exist some improved version for this special case of real polynomials ?
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1answer
74 views

Problems with basic algebra

I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
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1answer
115 views

Prove Using Complex Multiplication

Show the folowing general arctangent identity using complex multiplication, $\arctan\frac{1}{a-b} = \arctan\frac{1}{a} + \arctan\frac{b}{a^2-ab+1}$, for distinct real numbers $a$ and $b$.
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3answers
818 views

Complex number inequality.

If z and w are distinct complex numbers such that $|z| =|w| = r$, prove that $|\frac{1}{2}(z + w)| < r$.
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2answers
120 views

Computing with Cauchy Residue theorem

how do I calculate $$\operatorname{Res}\left(\frac{1}{z^2 \cdot \sin(z))}, 0\right)$$ What is the order of the pole? $3$?
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1answer
41 views

Nadirashvili surface (part 2)

The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure ...
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1answer
170 views

Converting to complex form

Write each of the given numbers in the form $a+bi$. a.) $e^{\frac{-i\pi}{4}}$ b.) ${\frac{e^{1+i3\pi}}{e^{-1+i\pi /2}}}$ c.) $e^{e^i}$ For a, I got $(\frac{\sqrt 2}{2} -\frac{\sqrt 2}{2}i)$, which ...
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1answer
56 views

Cyclotomic polynomial simplification

I try to work out what $\Phi_{12}(z)$ is: By the fundamental theorem of arithmetic: ...
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1answer
1k views

Solving systems of linear equations in the complex field.

I need to show that two systems of linear equations are equivalent, however this is over the complex field. How would I solve this? One of the systems is: $$\left\{\begin{align*} ...
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1answer
70 views

What is more compact equation of this relationship?

What is more compact equation of this relationship? $\sum |x_i|^2\sum |y_j|^2+\sum |x_j|^2\sum |y_i|^2-2|\sum x_i \overline y_i||\sum x_j \overline y_j|$ Remark: Euclidean space $\sum x_i^2\sum ...
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1answer
72 views

Find the Möbius transformation mapping $(i, 0, \infty)$ to $(0, \infty, -i)$, in precisely that order

I did this: Assuming my Möbius transformation is some $\omega$ in terms of $z$, I want to work out a formula that gives me: 1) $\omega = 0$ when $z = i$ 2) $\omega = \infty$ when $z = 0$ 3) $\omega ...
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1answer
105 views

question about solution of equation complex variable

A friend just told me that the equation $e^{z^2}=0$ has solution. Is it true? Thanks, Dan
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1answer
83 views

Let $f(z)=e^x + ie^{2y}$ where $z=x+iy$. Where does $f'(z)$ exist?

Let $f(z)=e^x + ie^{2y}$ where z=x+iy is a complex variable defined in the whole complex plane. a)Where does f'(z) exist? b) Where is f(z) analytic? Answer: a) I used the Cauchy Riemann to test ...
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2answers
694 views

Complex solutions of $\sin z = i \alpha \cos z$

I'm trying to solve the following question: Let $\alpha$ $\in [-1, 1]$ be a real number. Find all complex numbers $z$ that satisfy the equation: $\sin z = i \alpha \cos z$ This is what I've done ...
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2answers
81 views

Multiplying imagionary roots of a polynomial

I am trying to answer the following question: The roots of the quadratic equation $ax^2-16x+25$ are $2+mi$ and $2-mi$, where $m>0$. Compute the sum of $a+m$. Should the zeros of the equation ...
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1answer
130 views

exercise on complex numbers

Let $$f(z)=\frac{z-a}{z-b}$$ with $a,b\in D(0,r)$ and $r>0$. Let $$E=\{z\in\mathbb C: f(z)\notin N\}$$ $$N=\{Re(z)\leq 0;Im(z)=0\}$$ How can i find $E$ in terms of $r$?