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

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Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$

I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you. Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for ...
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77 views

What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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1answer
88 views

Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$

This is an old qualifying exam problem that I am working on. I would appreciate some help. Thank you. Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$ and ...
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1answer
40 views

Proof using de Moivre's Theorem

Let $z=\cos\theta + i\sin\theta$ Show that $$1+z = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2})$$ I don't even know how to start on this proof.
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23 views

An inequality on the real part of a square root

I have the following inequality: $\Re(k+z) \geq \Re \sqrt{(k+z)^2-4z}$ where $k$ is real and $z$ complex. Under what conditions on $k$ and $z$ is this inequality true? I suspect that it is true for ...
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1answer
61 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
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1answer
43 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...
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1answer
44 views

Imaginary unit $i$ is not a limit of a real Cauchy sequence

I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued ...
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2answers
43 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
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33 views

Let z != -1, which module is 1. Prove that z is presentable z = (1+ti)/(1-ti), where t is real number

Let z != 1, which module is 1. Prove that z is presentable in the following form: $$ z =\begin{align} \frac{1 + ti}{1 - ti} \end{align}$$ where t is a real number So, im guessing i have to ...
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1answer
19 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
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0answers
14 views

What wave does a “complex frequency” correspond to in the Fourier Transform?

The Fourier Transform takes a function $f$, you get another a function $g$. $g$ takes a complex frequency, and returns a sort of relative amplitude of a wave function in $f$. My question is how do you ...
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1answer
71 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
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36 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
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1answer
84 views

This integral is strange

$$ \int_{C_1}\frac{dz}{z}=\int_0^{2\pi}\frac{-R\sin{t}+iR\cos{t}}{R\cos{t}+iR\sin{t}}dt=\int_0^{2\pi}i\text{ }dt=2\pi i\tag{24.36} $$ Shouldn't it simply be $$\left[\ln(R \cos t + iR \sin ...
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51 views

What is the polar form of -6i?

The module of -6i is 6 (the square root of 36), but $ tan\theta = -6/0$, meaning that the polar form $ 6(cos\theta + isen\theta) $ is also indefinite?
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32 views

Comparison of two matrix multiplication operations.

I am comparing the below operations: $$ A=\begin{bmatrix} a & 0 & f & g \\ 0 & b & 0 & 0 \\ f & 0 & c & h \\ g & 0 & ...
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2answers
23 views

Re z/z continuous at z=0

How would I show that Re z/z is continuous at z=0? I know that the real value of a complex number equals the sum of the real number and its conjugate divided by two, but I'm not sure where to go when ...
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50 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
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Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
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Given that $x$ is a rational number, is $\sin(x\pi)$ always expressible through radicals?

This is a theory I just thought of and I am wondering if there is truth to it. Here is the logic that I am working upon: Using Euler's formula, you can deduce that $$ (-1)^x = ...
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24 views

What is the name for the operation of swapping the two components of a complex number (rectangular form)?

I wonder if there is a name for the operation of swapping the real and imaginary part of a complex number.
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61 views

Solve the recurrence relation $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$

This is a problem I was playing with that troubled me greatly. $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$ $f(1) = f(2) = 1$ $f(3) = 2$ So, the goal is to try and find a solution for f(n). I tried ...
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1answer
68 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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36 views

$\text{Im}(z)$ in equation

I'm having trouble with this equation: $$\text{Im}(-z+i) = (z+i)^2$$ After a bit of algebra i've gotten: $$1-\text{Im}(z) = z^2 + 2iz - 1$$ But i have no clue where to go from here, how do i get ...
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1answer
57 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
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1answer
34 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
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518 views

Extending the set of complex numbers

Mathematics as a science became richer when Cantor invented the real numbers. Then scientists wanted to solve equations which were not solvable in the real numbers so they invented the complex ...
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42 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
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1answer
22 views

The principle argument of the product of two complex numbers in the second quadrant

I would like some help to prove the following: Show that, if Re $z_1<0$, Im $z_1>0$, Re $z_2<0$ and Im $z_2 >0$, then Arg$(z_1z_2)=$Arg$(z_1)+$Arg$(z_2)-2\pi$. Thanks for any help in ...
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1answer
56 views

Finding the period of the solution to $y'(x) = y(x) \cdot cos(x + y(x))$ with Fourier transform; how to interpret complex result?

A question elsewhere on this site asks about detecting the frequency of oscillations in a system defined by differential equations. The equation is $y'(x) = y(x) \cdot cos(x + y(x))$. The solution ...
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1answer
35 views

Principle root of 3+4i

Is there a neat way of writing the principle root of 3+4i? I have an answer, but it is very ugly. Thanks for any help in advance.
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93 views

$\lim_{x\to 2} \, \sqrt{x-2}$

$$\lim_{x\to 2} \, \sqrt{x-2}$$ When you take the right hand limit for this expression, you get $0$. However, if you take the left hand side it gives an imaginary number. However, do you consider ...
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34 views

Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
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108 views

How to show that…, $A(C_1)=\{z:|z-1|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$?

Let $C_1=\{z:|z|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$ and $A(z)=z-1$. Define $A(C_1)$. How to show that $A(C_1)\neq\{z:|z-1|\leq 1, \arg(z)=\theta \in [\frac{3\pi}{4},\frac{5\pi}{4}] ...
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Contradiction with complex gaussians…

So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. The setup: Let $C$ be a complex constant, that is, ...
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Surjective complex map.

Show the map $(a + bi) \mapsto (a-bi)$ is surjective. Attempt: By definition, for every $(a - bi)$ in the complex set, there exists an $(a + bi)$ in the complex set such that $f[(a + bi)] = a - bi$. ...
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31 views

Simplifying $\exp {- i 2 \pi / N}$.

a and b are complex numbers and I know the equation below. $$X_{N} = a + e^{-i2\pi /N}*b$$ I wanted to simplify it. Here is what I've tried. I know $e^{-i\pi} = -1$ $X_{N} = a + \left ( e^{i\pi} ...
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Imaginary number in relativistic speeds

I am layman in field of mathematics but when I was reading about theory of special relativity I have come across speed limit of light and the book said that no one can cross that limit because ...
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2answers
56 views

Complex Numbers Exercise [closed]

If $a,b,c$ are complex numbers with $a+b+c=0$ and $\|a\|=\|b\|=\|c\| = r>0$ then prove that $$a^{2^n} + b^{2^n} + c^{2^n} = 0$$ Any ideas? Thanks!
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Find the roots of the simple equation?

x^{2}= 0 What are the roots? are they in complex plane, but how? Answer seems trivial in real numbers ain't it? Does this evolve a new system like it was with iota?
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30 views

Please refer book on complex numbers - especially covering equations of complex variables topic

I am searching for a good book to cover topics of complex numbers. Please refer book on complex numbers - especially covering equations of complex variables topic . Example : If $\alpha$ is a ...
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Cyclic sum inequality involving five numbers with modulus one and zero sum

When working on this MSE question, I was led to conjecture the following : If $z_1,z_2,z_3,z_4,z_5$ are five complex numbers with modulus $1$, such that $z_1+z_2+z_3+z_4+z_5=0$, then $$ ...
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locus of complex $z=(λ+3) + i\sqrt{3-λ^2}$

if $z=(λ+3) + i\sqrt{3-λ^2}$, for all real $λ$, then the locus of $z$ is ? Please help. Options are (A) circle (B) parabola (C) line (D) none of these
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Complex Number and Geometry

Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at ...
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1answer
49 views

Consider a quadratic equation $az^2+bz+c=0$ where a,b,c are complex numbers. Prove that the equation has one purely imaginary root is given …

Problem : Consider a quadratic equation $az^2+bz+c=0$ where a,b,c are complex numbers. Prove that the condition such that the equation has one purely imaginary root is given by ...
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175 views

True or False: $i^2 = -1$, $\mathbb{C} = \mathbb{R}^2$

The complex numbers are typically defined as the set of all ordered pairs $(x,y),$ where $x,y \in \mathbb{R},$ along with the usual operations of addition and multiplication (which I won't write out ...
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25 views

Finding roots of complex quadratic equation

I'm trying to solve for the following equation: $$|(1+50*i*x)^2|$$ I keep getting the form $$-2500x^2 + 100ix + 1 $$ when the problem needs to have the following form: $$2500x^2 + 1$$ What steps ...
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1answer
41 views

Prove that $\bar{P}_{\bar{z}}=P_z,\ (P_z,P_z)=(P_{\bar{z}},P_{\bar{z}})$ with $P_z=\dfrac{\partial{P}}{\partial{z}}$

I have a problem: For $P$ is a nonzero real valued homogeneous polynomial of degree $k$: $$P(z,\bar{z})=\sum_{j=1}^{k-1}a_jz^j\bar{z}^{k-j}$$ where $a_j \in \Bbb C,\ a_j=\bar{a}_{k-j}$. ...
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3answers
36 views

What does $\theta = \text{arg}(a,b)$ mean?

I have this equation where an angle is calculated using following formula: $$\theta = \text{arg}(C_1, C_2)$$ where $C_1, C_2$ are some numerical values. What exactly does it mean?