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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2answers
67 views

How do I solve equation $\bar{z} = |z|$ correctly?

I'm having troubles, finding how solution would look like for complex equation of the form $\bar{z} = |z|$. Taking $z = x + iy$, we get the following: $$x - iy = \sqrt{x^2 + y^2},$$ then raising it to ...
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0answers
24 views

A question about the log of a rational function

We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that : $$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\...
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4answers
77 views

Is the derivative of $x^2 + C$?

What is the derivative of $x^2$+C, except if C was set to the imaginary unit $i$? It wouldn't be possible to take, or would it simply be $2x$?
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0answers
26 views

Elements of $\mathbb Z [i]/\langle 1+4i\rangle$

What are the elements of $\mathbb Z [i]/\langle 1+4i\rangle$ ? I know that there are 17 elements as the norm of $1+4i$ is 17 but I can't manage to find them...
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1answer
87 views

Difference between Euler form and polar / trig form of a complex number

After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the ...
0
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1answer
33 views

Complex Number Systems involving trig.

I have to integrate from $0$ to $1$ in complex numbers the quantity $e^{-t} \cdot sin(2\pi t)$ I know what sign should look like if that $2\pi$ was not there, but it is, so do i just but it in for ...
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2answers
27 views

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$, suggestion: Reduce this inequality to $(|x|-|y|)^2 \geq0$ (z is a complex number. R stands for real part and Im stands for imaginary part) Approach: Let $...
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1answer
44 views

Prove that if $z_1*z_2$=0 then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers

Prove that if $z_1*z_2=0$ then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers by using the following property: $|z_1z_2|=|z_1||z_2|$ Approach: if $z_1*z_2=0$ then $$|...
6
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2answers
484 views

Identities derived from Euler's Identity

I am just learning about Euler's identity. When messing around with it, I am getting some unsettling identities, which, I believe, is probably due to my lack of application of certain rules. Starting ...
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1answer
32 views

$A$ is a square complex matrix. $A^k-A=0$ for some $k\geq 2$. Prove that $A$ is diagonalizable over $\mathbb C$

$p(x)=x^k-x=x(x^{k-1}-1)$ What I want to do is to say that $(x^{k-1}-1)=(x-z_1)(x-z_2)...(x-z_{k-1})$ and therefore A is diagonalizable (because of the distinct roots in the polynomial), but i'm not ...
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2answers
24 views

Problem based on amplitude of complex number

If the expression $(1 + ir)^3$ is of the form of $s(1 + i)$ for some real $s$ where $r$ is also real and then the value of $r$ can be $(A) \cot{\frac{\pi}{8}}$ $(B) \tan{\frac{\pi}{12}}$ $(C) \tan{\...
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3answers
37 views

The real or purely imaginary solutions of the equation, $z^3+iz-1=0$ are?

The number of real or purely imaginary solutions of the equation, $z^3+iz-1=0$ is? I substituted $x+iy=z$ and am getting two equations which seem impossible to solved.What would be the correct ...
2
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2answers
25 views

Question about triangle inequality

By factoring $z^4-4z^2+3$ into two quadratic factors an using the triangle inequality, show that if $z$ lies on the circle $|z|=2$ ($z$ is a complex number) then $$\left|\frac{1}{z^4-4z^2+3}\right| \...
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1answer
54 views

complex sequences

my series and sequence knowledge has gone a little rusty so I was wondering if you could help me on the right path here. The assignment is to calculate the sum of the series $(\frac{1}{8})^n e^{j(n{\...
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1answer
31 views

Find the 8th complex roots of $-81i$.

Find the 8th complex roots of $-81i$. So $i = \cos(\pi/2) + i\sin(\pi/2) = \cos(5\pi/2) + i\sin(5\pi/2)$ thus $z^8 = -81e^{(\pi + 4k\pi)/2 * i}$ So, $z = \sqrt{3} \cdot e^{i \cdot \frac{3\pi + 4k\...
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1answer
42 views

Complex Number Application in Isosceles Triangle Incentre

On the Argand plane $z_1, z_2$ and $z_3$ are respectively the vertices of an isosceles triangle ABC with AC = BC and equal angles are θ. If $z_4$ is the incentre of the triangle then prove that $(...
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3answers
37 views

What is the meaning of having a position vector $\vec r = z_1 \hat i +z_2 \hat j +z_3 \hat k$ where $z_1,z_2$ and $z_3$ are complex numbers?

What is the geometrical interpretation of this? I came across this as a solution to a second order differential equation when I was solving a problem including $m\vec a = f(\vec r)$. Thank you for ...
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2answers
542 views

How can I show that circles in the complex plane correspond to circles on the Riemann sphere? How about lines?

Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line. Here we are defining $ \...
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1answer
71 views

What does $\sqrt{2i}$ imply in a question?

While doing a certain problem based on complex numbers I faced this doubt. When $\sqrt{2i}$ is mentioned in a question should I take it's value as $(1+i)$ or both $(1+i)$ and $(-1-i)$ ?I mean should ...
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6answers
3k views

Applications of Complex Numbers

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start ...
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2answers
20 views

Considering the complex number $z = m+i$ for which values of $m$ do we have $ \left|\overline{z}+\frac{2}{z}\right| \ge 1 $

Good evening to everyone. I have the following problem that I tried to solve but my mathematical instinct tells me that I didn't solve it right: Considering the complex number $z = m+i$ for which ...
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0answers
17 views

Preposition about the Entries of the Product of Markov Matrices.

Definition: A Markov matrix is an $n \times n$ complex matrix with the sum of the elements in every column equal to 1. My task is to prove that: If A, B are Markov matrices such that $|a_{ij}|\leq1$ ...
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2answers
41 views

Tangent and Circle in Complex Plane

Question:- Three points represented by the complex numbers $a,b$ and $c$ lie on a circle with center $O$ and radius $r$. The tangent at $c$ cuts the chord joining the points $a$ and $b$ at $z$. ...
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3answers
1k views

Example of a complex transcendental number?

Researching transcendental numbers I have only come across ones with a transcendental real part. I can't think of any which are pure imaginary or are not based on a real transendental number, t, of ...
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2answers
78 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
5
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1answer
69 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
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0answers
17 views

find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
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4answers
200 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
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3answers
164 views

Subtracting roots of unity. Specifically $\omega^3 - \omega^2$

This is question that came up in one of the past papers I have been doing for my exams. Its says that if $\omega=\cos(\pi/5)+i\sin(\pi/5)$. What is $\omega^3-\omega^2$. I can find $\omega^3$ and $\...
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3answers
45 views

Matrix and scalar multiplication

Say we have the following variables: A, a matrix that is nxn in size containing complex numbers B, a matrix that is also nxn in size containing complex numbers x, a scalar If you multiply, does it ...
1
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1answer
26 views

multiplying 3 variable a ,bi, cj

I know how to do this $A × B = ( a + bi ) × ( c + di )$, but I don't understand how to do this $ A × B =( a + bi +cj) × ( d + ei +fj) =? $ I'm not sure how to group them or what to do with the i ...
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4answers
219 views

Solve $z^5 +32 =0$

Solve $z^5 +32 =0$ My attempt : $$z^5 = -32$$ Multiply the powers on both sides by $\frac{1}{5}$ we get $$z = 2 * (-1)^\frac{1}{5}$$ Now I'm stuck at this step I don't know how to ...
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0answers
14 views

Start and endpoint of line, creating arrow heads [on hold]

I have a start point(5.6,4) and an endpoint (6.1,3.15) I want to make an arrow head at the start point that is an equilateral triangle(60 degrees) with a length of .1. How can I accomplish this? ...
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2answers
27 views

Squareroot of complex number to the square $\sqrt{z^2}$

I have to calculate $\sqrt{z^2}$ an I am confused about how to procede. I thought about introducing $z=|z|\exp(i\phi+2\pi k) \implies z^2=|z|^2\exp(2i\phi+4\pi k)$. Hence, $$\sqrt{z^2}=\sqrt{|z|^2\...
2
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1answer
22 views

Mutually orthogonal vectors in a complex vector space?

Consider a Matrix $A \in \mathbb C^{m \times n}$, $m<n$ which is build by vectors like $$ A = \begin{pmatrix} | & | & & | \\ \vec a_1 & \vec a_2 & \cdots & \vec a_n \\ | &...
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1answer
140 views

Complex transcendentals not known in component form?

Are there any transcendentals whose real or imaginary components have not been found in exact form?
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1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
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0answers
61 views

Spec$(R)$ a scheme of finite type over $\mathbb{C} \implies R$ is a finitely generated algebra over $\mathbb{C}$.

Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra. Since $(\text{Spec}(R), \tilde{R})$ ...
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4answers
646 views

Why are complex numbers considered to be numbers?

I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that ...
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3answers
78 views

When are complex conjugates of solutions not also solutions?

I've heard that for "normal" equations (e.g. $3x^2-2x=0$), if $(a+bi)$ is a solution then $(a-bi)$ will be a solution as well. This is because, if we define $i$ in terms of $i^2=-1$ then we might as ...
30
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7answers
3k views

Can a complex number be prime?

I've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime.
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2answers
49 views

Complex Numbers: A Basic Manipulation

$$ z = e^{i \phi} \tan \Bigg (\frac{\theta}{2} \Bigg). $$ What is, $$ \frac{|z|}{1 + |z|^2} $$
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2answers
36 views

How to separate real and imaginary parts of an expression

i am taking a course on complex numbers and I need to know how to separate the real and imaginary parts of a trigonometric expression like 1) $$\cos^{-1}(ix)$$ 2) $$\sin^{-1}(e^{i\theta})$$
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2answers
997 views

Clarification on “Every polynomial function of degree $\ge1$ has at least $1$ zero in the complex number system.”

The Fundamental Theorem of Algebra says "Every polynomial function of degree $\ge1$ has at least $1$ zero in the complex number system." My question is, where do the rest of the zeroes of the ...
5
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3answers
2k views

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
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7answers
2k views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I wrong?...
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3answers
65 views

If $|z|=\sqrt{a^2+b^2}$, then what is $z$?

Perhaps I’m having some difficulty understanding the complex plane. Say you have a complex number $z=a+bi$, where $a$ is the real part and $b$ is the imaginary part. Why do you plot the real part on ...
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3answers
82 views

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represent an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represent an ellipse?, with $A,B \in \mathbb C$ How can it be described?
5
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2answers
100 views

Why doesn't $e^x$ have an inverse in the complex plane?

Why doesn't $e^x$ have an inverse in the complex plane? Can someone please clarify it?
3
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2answers
82 views

Prove that there does not exist a $n$-regular polygon $(n\ge 4)$, such that its sides and diagonals are all integers.

Prove that there does not exist a $n$-regular polygon $(n≥4)$, such that its sides and diagonals are all integers. Maybe a famous problem, but I don't know how to solve that.