Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
-2
votes
0answers
20 views
Compactness of $S\subseteq \mathbb{C}^2$
Let $M$ is a positive operator on a complex Hilbert space $F$. My goal is to show that the following set
$$S_M=\{x\in F\,;\;\langle Mx,x \rangle=1\},$$
is not always compact.
Note that by a positive ...
0
votes
2answers
31 views
Why does $i = 0$ or $\tau = 0$ following this logic?
Following this logic, $i$ appears to equal $0$:
$$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$
$$e^{i\tau} = \cos(\tau) + i\sin(\tau)$$
$$e^{i\tau} = 1 + i(0)$$
$$e^{i\tau} = 1$$
$$e^{i\tau} = e^0$$
...
0
votes
0answers
15 views
Sketch the region
$ Re(z)+Im(z)= \frac{1}{Re(z)-Im(z)}$
Replace Re(z) with x and Im(z) with y to get:
$ x+y= \frac{1}{x-y}$
Then:
$ (x+y)(x-y)=1 $
$ x^2-y^2 =1 $
Assuming i did my math properly all i have to do is ...
0
votes
3answers
47 views
Defining the complex numbers as the algebraic closure of the real numers [on hold]
It is of course possible to define the complex numbers as a quotient ring of real polynomials: https://math.stackexchange.com/a/1083130/359302
But is it possible to prove all of their properties - ...
0
votes
2answers
35 views
Explaining the argument formula
I am beginning my study of Complex Analysis, and I stumbled on the following definition on how to compute the argument:
$\varphi = \arg(z) =
\begin{cases}
\arctan(\frac{y}{x}) & \mbox{if } x >...
1
vote
2answers
48 views
Find all solutions in $\mathbb C$ for $z^4 = 1$
To start, I write the equation in polar form:
$$|z|^4(cos^4\theta + isin^4\theta) = 1(1 + 0i)$$
Next, I want to solve for $\theta$:
$$cos4\theta = 1 \textrm{ and } sin4\theta = 0$$
$$4\theta = cos^{-...
0
votes
2answers
78 views
$\frac{1}{\sqrt{-1}}=\sqrt{-1}$? [duplicate]
I have trouble to comprehend what my mistake is in the following calculation:
If we set $\sqrt{-1}$ to be the new number with the property that $(\sqrt{-1})^2 = -1$ then I can write $$\frac{1}{\sqrt{-...
1
vote
0answers
16 views
Comparing four ways to visualize complex functions
Among many others there are these four ways to visualize a complex functions $f(z)$:
Choose
a straight line going through the origin
a circle around the origin
For each $z$ on this curve draw a ...
0
votes
0answers
24 views
Using Euler Eqns to separate a quadratic into real and imaginary components
I'm working on a research paper on Delay Differential Equations and have arrived at the same characteristic equation for the linearised system:
\begin{align}
\lambda^2+p\lambda+r+(s\lambda+q)e^{-\...
0
votes
0answers
19 views
Complex even and odd polynomials with real coefficients sharing a zero
It's true that if an $f:\mathbb{C} \to \mathbb{C}$ is an odd polynomial and $g:\mathbb{C} \to \mathbb{C}$ is an even polynomial are polynomials with reals coeficients and they share a root, then that ...
0
votes
1answer
32 views
Proving the continuity of a complex valued function
So I am given the following function
$$f(z)\begin{cases}
z & |z|\leq 1 \\
|z|^2 & |z| > 1
\end{cases}$$
and I want to find all points of continuity. I know that for $|z| &...
2
votes
2answers
44 views
How the way of finding roots of a complex number works
I would like to describe my doubt with a question
Finding the sixth roots of
$$ 16 - 16\sqrt{3}i $$
One can solve this question by obtaining
$$ z^6 = r^6\exp(i\theta) = 32\exp((-\pi/3 + 2\pi k)/6)$$...
1
vote
0answers
18 views
Prove that $z_1,z_2,z_3$ with equal, non-zero modulus, are vertices of an equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $
Prove that $z_1,z_2,z_3 \in \Bbb C$ , distinct, with equal, non-zero modulus, are vertices of equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $
I tried dividing by $z_3|z_3|\...
3
votes
2answers
51 views
Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition
I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
0
votes
1answer
43 views
Modulus of a complex number depending on a parameter
How to solve |2 + t + t*i|$^2$? I know the formula for z = x + iy, then |z| = $\sqrt{x^2 + y^2}$
0
votes
2answers
32 views
Complex number vs Complex exponential
I know that a complex number is a point in 2D plane. I wonder how to describe what is a complex exponential?
0
votes
1answer
34 views
Locus of complex number.
I have a locus of points $Z$ that satisfy equation: $Z = a + bt + ct^2$ where $a,b,c \in \mathbb{C}$, $t \in \mathbb{R}$ is a parameter and $\frac{b}{c} \in \mathbb{R}$, but I don't know to proceed ...
0
votes
1answer
41 views
Complex roots of the polynomial
In C[x] one of the roots of the polynomial $V(x)$ is $X1=-1+i$. Find the remaining roots of $V(x)$.
I'm so sorry for the picture, but I need extra help with it. Hope you understand me.
Probably my ...
3
votes
2answers
76 views
Proving that |z|=1. [duplicate]
I am trying to prove that
If $z\in \mathbb{C}-\mathbb{R}$ such that $\frac{z^2+z+1}{z^2-z+1}\in \mathbb{R}$. Show that $|z|=1$.
1 method , through which I approached this problem is to assume $z=a+ib$...
1
vote
0answers
32 views
unusual “distance” in complex space
I've encountered a notion of distance (their usage) in the field of ecology that raises an eyebrow, and would like to know if it's used elsewhere (if it makes any sense at all).
If you diagonalize a ...
1
vote
1answer
45 views
Find the image of complex function $w=z^2+\frac{1}{z^2}$
I need to find the image $\Gamma_w$ of complex function $w=z^2+\frac{1}{z^2}$, with the domain $$\Gamma_z = \{ z \in \mathbb{C}\ |\ 0 \leq \arg z \leq \pi/2 \wedge |z| \geq 1\}.$$
Using $z=re^{i\phi}$...
0
votes
1answer
54 views
Conformal map from upper half plane to the slit unit disk
I am trying to find a conformal map from $\mathbb{H}=\{z:Im(z)>0\}$ onto $\mathbb{D}$ take away $(-1,0]$, where $\mathbb{D}=\{z:|z|<1\}$ (the slit unit disk).
I have found that $f(z)=\frac{z-i}...
1
vote
1answer
35 views
Finding Taylor series for a complex logarithm branch
The Problem:
Find the Taylor series for the logarithm branch $0<\arg(z)<2 π$ in powers of $z+2$
My resolution:
The method I've used to come up with an answer was integration of a known series. ...
1
vote
1answer
35 views
How to write $e^{\pi+i}$ as $a+bi$
How can I find the $a+bi$ form of the number $e^{\pi+i}$?
Normally, it is $e^{i \pi} = \cos(\pi) + i \sin(\pi)$ but in this case, I don't find any clue.
3
votes
2answers
40 views
What can be said about complex numbers $z_1, z_2, z_3$ if $\frac{z_1 - z_3}{z_2 - z_3}$ is real?
$z_1, z_2, z_3 \in \mathbb{C}$
$$\frac{z_1 - z_3}{z_2 - z_3} \in \mathbb{R}$$
The only idea I'm coming up with is that either $ \operatorname{Im}(z_1) = \operatorname{Im}(z_2) = \operatorname{Im}(...
0
votes
2answers
43 views
Can you prove the Pythagorean theorem using only algebra with complex numbers or quaternions? [on hold]
Is it possible to prove $|a+bi|=\sqrt{|a|^2+|bi|^2}$, where $|x|$ is the distance of $x$ from zero, using only algebraic manipulation with complex numbers, or maybe also quaternions if needed?
1
vote
0answers
10 views
Notion of line on a complex affine space
$\newcommand{C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\i}{\hspace{0.1em}\mathrm{i}} \newcommand{\R}{\mathbb{R}}$
Let $V = (\C, \C, +, \cdot)$ be the one-dimensional complex vector space ...
0
votes
0answers
8 views
Can we think this reverse reading as generalization metric tensor 'action'?
Metric tensor is so defined
A metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and ...
0
votes
0answers
5 views
The problem of the Chebyshev approximation in complex sense
Let there be given the set of points H of the real line,and the real functions $f(x)$ and $F(x; \lambda_1 ,\lambda_2 .... ,\lambda_n)$ of the real variable $x$, which are bounded on this set. We have ...
1
vote
2answers
34 views
Why is $z=1$ not a branch point of the function $w=f(z)=z^{1/2}$?
Consider the function $w=f(z)=z^{1/2}$ and the point $z=1$ on the $z$-plane. Next consider a closed circular loop of radius $2$ about the point $z=1$ so that $w=1$. As we go around $z=1$ along this ...
2
votes
1answer
65 views
$f(z) = \frac{z-a}{1 - z\bar{a}}$ interpretation?
I recently started a complex-analysis course and our teacher proposed this exercise to us:
Given $a, z \in \mathbb{C}$, consider the complex function
$$ f(z) = \frac{z-a}{1 - z\bar{a}}.$$
I've ...
2
votes
1answer
25 views
Elimination of zeros of an entire function
Let be $f$ an entire function and $a_1,a_2,...,a_N$ the zeros of $f$ (i.e. $f(a_k)=0$). A $g$ function as: $$g(z)=\begin{cases} \frac{f(z)}{(z-a_1)(z-a_2)...(z-a_N)}, \qquad \text{for }z\neq a_k, \\ \...
4
votes
2answers
96 views
Do we know the limit of $\sum\limits_{k=1}^{n}\frac{1}{(a i k+b)^2}$? [closed]
I am not sure if the closed- form of this limit is known ($i$ is the imaginary one):
$$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{(a i k+b)^2}$$
If it's not, then yay, I found it. But I think it's very ...
0
votes
4answers
27 views
what is the amplitude and argument of given z= 1-cos(9$\pi$/10)-i*sin(9$\pi$/10)
What is the amplitude and argument of the given complex number below?
$ z= 1-\cos(9\pi/10)-i \cdot\sin(9\pi/10)$
I have tried this a few times but the answer won't match
the answer to this problem ...
0
votes
1answer
38 views
Determine if the given complex matrices are similar.
Let $a,b,c,d,e,f$ be complex numbers.
$$A = \left( \begin{array} { c c c c } { 2 } & { a } & { b } & { c } \\ { 0 } & { 2 } & { d } & { e } \\ { 0 } & { 0 } & { 2 } &...
0
votes
1answer
17 views
Need Help with Complex Equation and finding all Zs
I'm preparing for an exam and I'm facing some troubles with complex numbers, so any help would be much much appreciated! I've been re-reading the chapter like 10 times by now and I just can't figure ...
0
votes
3answers
54 views
Triangle Inequality Proof: Show that:$||z_1|-|z_2||\leq |z_1+z_2|$ [duplicate]
I've been puzzled by this deceptively simple-looking complex number modulus inequality proof.
Show that:$$||z_1|-|z_2||\leq |z_1+z_2|.$$ State the condition for the equality to hold.
I tried
$\...
1
vote
1answer
39 views
How to prove the inequality $|e^z-1| \geq |z|/2$ for sufficiently small $|z|$?
How to prove the inequality $|e^z-1| \geq |z|/2$ for sufficiently small $|z|$?
I was thinking about the Taylor series for $e^z$, but I have no idea.
Any answers will be greatly appreciated.
1
vote
2answers
30 views
If $z\in\mathbb{C}$ satisfying $|z-1|=1$ then prove $\arg(z-1)=2\arg z$
If $z\in\mathbb{C}$ satisfying $|z-1|=1$ then which of the following is correct
\begin{align}
&a)\quad \arg(z-1)=2\arg z\\
&b)\quad 2\arg z=2/3.\arg(z^2-z)\\
&c)\quad \arg(z-1)=\arg(z+1)...
1
vote
4answers
41 views
How do I solve $z^2+(3+4i)z-1+5i=0$
Right, I asked yesterday about the explanation for the roots of quadratic equations, now I'm trying to apply these concepts.
As stated in the title, we start with:
$$z^2+(3+4i)z-1+5i=0$$
If we ...
0
votes
1answer
36 views
What is meant by “rational operation”?
So I am attending a course in complex analysis and came across the following statement pretty early in the book:
Let $R(a,b,c,\dots)$ stand for any rational operation applied to the complex numbers $...
0
votes
0answers
11 views
Velocity vector imaginary to real
So I have this equation:
$V = \frac{A}{\rho*c}*\frac{exp(i*(\omega*t-k*r))}{\rho*c}*(1+\frac{1}{i*k*r})\hat{x}$
The i in the equation makes it imaginary. I am trying to solve for the real portions ...
0
votes
2answers
36 views
what is the value of z/w?
Let $|w|=3$ come out of the origin and and let $|z|=2$ and also come out of the origin. Both lines are in the second quadrant and form an angle of $\frac{\pi}{3}$ degrees. What is the value of $\frac ...
0
votes
2answers
40 views
Find the polar form of $12 + 5i$
Polar form: $\vert z \vert \big(\cos\theta + i\sin\theta \big)$
$$\begin{aligned}z^2 &= \vert 12^2 + 5^2 \vert\\
z &= \vert 13 \vert \\
\arctan \frac{5}{12} &= 22.61^\circ\\
z &= |13| ...
1
vote
3answers
34 views
Help with solution to complex quadratic equation
I'm learning about solving complex quadratic equations, in the examle I'm following, we start with:
$$z^2+4iz-(7+4i)=0$$
This can be rewritten as:
$$(z+2i)^2-(3+4i)=0$$
The square can be rewritten ...
1
vote
4answers
42 views
Writing a number in polar form (help converting $\theta$ to $\pi$)
Write $w = \sqrt{3} - i$ in polar form.
How is $\theta = \frac{-1}{\sqrt3} \textrm{ converted into } \frac{-\pi}{6}$? I understand that $w$ lies in the fourth quadrant of the unit circle, but ...
2
votes
1answer
31 views
Existence of diagonal matrix that transforms invertible matrix in unitary
This questions arises from the answer to this previous question, which leaded me to 'relax' the statement that I wanted to prove.
Suppose that a matrix $M\in\mathrm{SL}_2(\mathbb{C})\setminus\{\pm I\}...
1
vote
0answers
45 views
Using dual complex numbers for combined rotation and translation
Dual quaternions may be used to perform combined rotations and translations in a single dual quaternion product operation.
Translation is performed by placing the displacement, $d$ in vector of the ...
0
votes
2answers
32 views
How to factorize equations in terms of $w$?
Like we can factorize $x^2 + x + 1$ as (x+$w$)(x+$w^2$).
Where $w$ is cube root of unity
1
vote
0answers
66 views
Proof: For every $a \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha + \beta = 0$
I am trying to prove the following:
For every $\alpha \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha + \beta = 0$.
There are two steps to this proof:
I first ...
