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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$ z^n = a_n + b_ni $ Show that $ b_{n+2} - 2b_{n+1} + 5b_n = 0 $ (complex numbers)

$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$ How can I solve this? Thank you! EDIT: Or ...
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1answer
27 views

Question about proof of triangle inequality in $\mathbb{C}$

I am having some trouble with one inequality used in the proof of the triangle inequality in $\mathbb{C}$. The main issue is realizing that for $z,w \in \mathbb{C}$, we have that $2 \cdot Re(z\...
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1answer
21 views

find the laurent series using z=w+1

Here is the question that my books is asking Find the same Laurent series for $f(z)=1/(z(z-1)^2)$ center at $c=1$ by using the following procedure. Set $z=w+1$, expand the resulting function in ...
2
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0answers
44 views

Heegner Prime visualizations

The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ring of integers $\textbf{Q}(\sqrt{-d})$ have unique factorizations. 1 gives the Gaussian integers. 3 gives the Eisenstein integers. 7 ...
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2answers
88 views

On finding the complex number satisfying the given conditions

Question:- Find all the complex numbers $z$ for which $\arg\left(\dfrac{3z-6-3i}{2z-8-6i}\right)=\dfrac{\pi}{4}$ and $|z-3+i|=3$ My solution:- $\begin{equation} \arg\left(\dfrac{3z-6-3i}{2z-8-6i}\...
1
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4answers
86 views

Solve $z^2+i\bar{z} = 0$

Need to solve: $$z^2+i\bar{z} = 0$$ I have tried to use the same method for the other exercise here: Solve $z^2+iz=0$ but I do not know how to manage the $\bar{z}$ Any help?
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2answers
46 views

Is it possible to calculate imaginary exponents without trig functions?

If I have a problem such as $2^i$, I would use the rules: $$ e^{ix} = \cos{x} + i\sin{x} \\ b^n = e^{n\ln{b}} $$ Applying this to the example $2^i$, I would let $x=\ln{2}$: $$ e^{i\ln{2}} = \cos{(\...
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4answers
57 views

Solve $z^2+iz=0$

I need to solve $$z^2+iz=0$$. I need to explicit trigonometric and algebric form. Since there is a i in the equation I'm not able to start with $z=\rho( \cos\theta ...
3
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3answers
104 views

On the proof that $\left|\frac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$

Question: Prove that $\left|\dfrac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$ if $|z_1|\lt1$, $ |z_2|\lt 1$ My solution: I had no idea how to go about this one so instead I started simplifying the ...
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1answer
32 views

Cauchy riemann and analytic functions

At what points, if any, is $f(z)$ analytic? $f(z)=(2x+y−x^2y)+i(3+2y−xy^2).$ Please help, very confused.. I know how to compare the C-R equations and know how to find them. One set yields the unit ...
3
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0answers
42 views

When is $(2cis\frac{2\pi}{3})^n$ real?

When is $(2cis\frac{2\pi}{3})^n$ real? Using de Moivre's theorem: $$(2cis\frac{2\pi}{3})^n = 2^ncos(\frac{2\pi}{3}n) + i2^nsin(\frac{2\pi}{3}n)$$ $$\therefore sin(\frac{2\pi}{3}n) = 0 = sin(0), ...
5
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2answers
71 views

Real roots of $z^2+\alpha z + \beta=0$

Question:- If equation $z^2+\alpha z + \beta=0$ has a real root, prove that $$(\alpha\bar{\beta}-\beta\bar{\alpha})(\bar{\alpha}-\alpha)=(\beta-\bar{\beta})^2$$ I tried goofing around with the ...
2
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3answers
176 views

Hi! Just wondering if any one can help me out with this roots question? [closed]

(i). Factorise $z^2 - 5z + 6$ and hence, solve the equation $ z^2 - 5z + 6 = 0$ (ii). Show that $z^2 - 5z + 6$ is a factor of $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$. (iii). Find the three roots ...
1
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2answers
71 views

To find the solution of the equation $2\left|z \right|-4az+1+ia=0$

Question:- For every real number $a \ge 0$, find all the complex numbers $z$, satisfying the equation $2\left|z \right|-4az+1+ia=0$ Attempt at a solution:- Let $z=x+iy$, then the equation $2\left|...
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1answer
19 views

Calculate, simplify and expand exponents with complex numbers

Can we somehow calculate $a^z$ where z is a complex number ? Does normal exponent rules like : $$a^b\cdot a^c=a^{b+c}$$ Still work when complex numbers are in the exponent ? For example, do these ...
1
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1answer
74 views

On the solution of the equation $z+ \alpha \left| z-1\right| + 2i = 0$

Question:- Find the range of real number $\alpha$ for which the equation $z+ \alpha \left| z-1\right| + 2i = 0$; $z=x+iy$ has a solution. Also find the solution. Attempt at a solution:- On ...
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3answers
81 views

When we say that $x^2$ = -4, would x = $\pm$2i or 2i?

I saw a problem on Brilliant.org recently that sparked quite a heated discussion. The question was: Compute $\sqrt{-4} * \sqrt{-9}$. Now, If $\sqrt{-4}$ = 2i, the answer can only be -6. However, if $\...
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2answers
29 views

About matrix diagonalization in C from the characteristic polynomial.

Ok the excercise is: You have one characteristic polynomial, it's: $\lambda^4 + \lambda^2$ Find two matrixes with this polynomial, one of them diagolalizable in C and the other one not. so the ...
3
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2answers
56 views

solutions of $\bar z = |z-2\Im(z)|^2$.

I need to find all the solutions of $\bar z = |z-2\Im(z)|^2$. I know that $z=x+iy$ and $\bar z=x-iy$ and then $2\Im(z)=2y$. But can someone show the algebra for what I do next?
4
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2answers
73 views

A generalization of holomorphic functions

Let's fix a matrix $A\in M_{2}(\mathbb{R})$. Assume that the following vector space of smooth functions is closed under complex multiplication: $$\mathcal{S}_{A}=\{f:\mathbb{C}\to \mathbb{C}\...
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2answers
68 views

Square root of $i$

What's my error ? $i^1$ means rotation of $90°$ in anticlockwise manner from positive real axis. So $i^{1/2}$ means rotation of $45°$. So square root of $i$ must have both part positive (real and ...
1
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2answers
28 views

Complex matrix 2x2 eigenvalues determination.

So i have a matrix A: \begin{pmatrix} i & i\bar i \\ 1-i &\bar i \end{pmatrix} How can i calculate the eigenvalues of a complex matrix like this? I already know how to do if the matrix is ...
0
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2answers
29 views

Find the $n$-th power of complex number

Let $z=1+2i$ be a complex number. Prove that for any $n \in \mathbb{N}^*$, the number $z^n$ has the following form: $a_n+ib_n$, with $a_n,b_n \in \mathbb{Z}$. I guess the solution lies in the ...
3
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1answer
131 views

show that $\prod_{k=1}^{n-1}\left(2\cot{\frac{\pi}{n}}-\cot{\frac{k\pi}{n}}+i\right)$ is purely imaginary number

Show that $$\prod_{k=1}^{n-1}\left(2\cot{\dfrac{\pi}{n}}-\cot{\dfrac{k\pi}{n}}+i\right)$$ is purely imaginary number where $i^2=-1$ where $n=2$ it is clear $$2\cot{\dfrac{\pi}{n}}-\cot{\dfrac{k\...
2
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1answer
42 views

Adjoin complex numbers to an arbitrary field? [closed]

This is probably nonsense but I'm throwing it out there. I don't think I can even explain the question very well: Has anyone seen bizarre things such as adjoining, say $i$ or $\pi$, to say a finite ...
2
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1answer
78 views

Imaginary Golden Ratio

While playing with the results of defining a new operation, I came across a number of interesting properties with little literature surrounding it; the link to my original post is here: Finding ...
4
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1answer
57 views

n-th roots of unity summing to $0$

Let $\zeta = e^{2\pi i/n}$ be an $n$-th root of unity, and let $S = \{\zeta^m|m=0,1,\ldots,n-1\}$ be the corresponding sets of all $n$-th roots of unity. Let $k \leq z$. Let $C \subseteq S$ such ...
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3answers
34 views

Localization of roots of complex quadratic equations

Let $a,b,c\in\mathbb C\setminus\{0\}$ be complex numbers such that $$b^2-4ac \neq 0.$$ We consider the equation $$ax^2+bx+c=0.$$ I am interested in general statements about the roots of this equation ...
0
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3answers
45 views

Complex multiplication definition

$$\left(a,b\right)\left(c,d\right)=\left(ac-bd,ad+bc\right)$$ I'm in a book dealing with quaternions and it says the above is the definition of multiplication for complex numbers. Can someone show ...
5
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0answers
42 views

Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers [closed]

As so far as usage in elementary number theory goes, what is the difference between the natural numbers, the integers, the rational numbers, the complex numbers, and the Gaussian integers?
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2answers
47 views

Inequality with complex root and positive imaginary part

Let $z$ be a complex number with $\mathrm{Im}(z)>0$, and we consider $$w:=\frac{-z+\sqrt{z^2-4}}{2}.$$ It is written that "we take the square root so that $\mathrm{Im}(w)>0".$ I want to prove ...
3
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1answer
68 views

How can I solve $y''=\frac{a}{y^2}$ where a is a (positive) constant?

Actually, I found out a way to solve that, but I can't get rid of complex numbers. And it does not make sense when it comes to complex numbers as the original question that involves this differential ...
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2answers
54 views

How is $\cos(x)={e^{jx}+e^{-jx}\over 2}$? [closed]

How to prove the following equation? What is proof for $$\cos(x)=\dfrac{e^{jx}+e^{-jx}}{ 2} \qquad \qquad j=\sqrt{-1}$$?
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1answer
40 views

How to find relations between the roots of a fourth degree polynomial which has only complex roots?

Let f(x) is a fourth degree polynomial such that $f(x) = x^4+x^3+x^2+x+1$. Let p be a root of f(x). Then which one of the following cannot be root of f(x): $p^2$ $p^3$ $p^4$ $p^5$ I know f(x) has ...
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2answers
36 views

How to turn the reflection about $y=x$ into a rotation.

If we reflect $(x,y)$ about $y=x$ then we get $(y,x)$. And because $x^2+y^2=y^2+x^2$ this can also be represented by a rotation. Using this we get: $$(x,y)•(y,x)=2xy=(x^2+y^2)\cos (\theta)$$ Hence ...
0
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1answer
45 views

Value of $\theta$

Find the roots of the equation: $$z^4 + 4=0$$ According to De Moivre's theorem: $$w_k = (r)^{\frac{1}{4}}e^{i\phi_k}$$ $$\phi_k = \frac{\theta + 2 \pi k}{4}$$ Since $$z=(-4)^{\frac{1}{4}}$$ $$r = ...
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0answers
23 views

Eigendecomposition of a complex symmetric matrix and the same with a shift on the diagonal

Suppose that the matrix $A\in \mathbb{C}^{n\times n}$ is complex symmetric matrix, thus is fulfills $A=A^T$ but also $A\neq A^H$, thus the matrix is not hermitian, but complex symmetric. Let's ...
2
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2answers
86 views

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the ...
2
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3answers
47 views

How to simplify $\Re\left[\sqrt2 \tan^{-1} {x\over \sqrt i}\right]$?

While solving $$\int \frac{x^2+1}{x^4+1}\,dx,$$ I tried to use partial fractions in the denominator by writing $x^4+1=(x^2+i)(x^2-i)$ And then I got $\Re\left[\sqrt2 \tan^{-1}{x\over \sqrt i}\right]$. ...
1
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1answer
35 views

Primitive 18-th root of unity problem involving congruences.

I have some doubts about this following problem, if you can please try to answer the congruence step: Let $ \omega$ be a primitive 18-th root of unity. Find $ n \in \mathbb Z$ such that: $ \omega^n =...
0
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1answer
21 views

Second order system - find -3dB frequencies and magnitude response analytically

Let's take some simple second-order system like $H(s) = \frac{j\omega T}{(1+ j \omega T)^2} $. I know that the magnitude response is simply the absolute of the function and the -3dB frequencies can be ...
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0answers
41 views

Prove or disprove: $\lambda\in\mathbb C$ is an eigenvalue of A $\implies\overline\lambda$ is an eigenvalue of A

I'm pretty sure it's true because every time I calculated eigenvalues for a complex matrix it eventually boiled down to solving a quadratic formula that gave me a conjugated pair, but not sure how to ...
3
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3answers
102 views

How can I calculate $(1+i)^{5404}$?

I saw a pattern while evaluating some other powers of similar complex number so I tried to calculate the above question by expanding it, please tell me if it is correct...? $(1+i)^{2} = 2i$ $(1+i)^{...
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1answer
25 views

Using the picture write down all values of $\sqrt[12]{z}$ and then find the main value of that number $z$.

Problem: Let $1+i\sqrt 3$ be one root of $\sqrt[12]{z}$. Display that number in complex plane and then, in that plane, display other roots of number $\sqrt[12]{z}$. Using the picture write down all ...
2
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1answer
120 views

Where does this equation come from? [duplicate]

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?
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2answers
52 views

Real Numbers Raised to Imaginary Powers? [closed]

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ...
1
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1answer
27 views

locus of complex number 2

Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the ...
0
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2answers
51 views

If $ f $ has pole at $0$ then show that $e^f$ can't have pole at $0$.

i am trying to show that if $ f $ has a pole at $0$ then $ e^f $ can't have removable singularity at $0$ ? I tried to show that but i have a problem . I assume that $e^f$ has removable singularity ...
0
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1answer
65 views

Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?

Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\...
0
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0answers
25 views

Circle and line construction of a compex number $z\in\mathbb C$

Let $C\subseteq\mathbb C$ be the field of constructible complex numbers; that is, it includes only the elements $z\in\mathbb C$ which can be constructed with circles and lines. The field $E\subseteq \...