Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
11,224 questions
1
vote
7answers
40 views
Difference between $z^2$ and $|z|^2$
Are $z^2$ and $|z|^2$ same?
Where $z$ is a complex number.
If imaginary part of $z$ is zero, then surely we can say they are both are same. What about if imaginary part of $z$ is non-zero?
-1
votes
1answer
30 views
verifying an equality
This is a detailed version of my question posted yesterday. I have an expression
$\mathcal{Im}[RT^*e^{-2ip}]=|T|^2$sin $p$, where $R=Ae^{ip}+Be^{-ip}$ and $p$ is a real number.
This ultimately should ...
2
votes
3answers
36 views
Complex number question - spurious solutions…
$z_1 = 2 + 3i$ and $z_2 = 3 - 4i$
The complex number $z = x + iy$ is such that $\frac{z + z_1}{2z - z_2} = 1$.
Find the value of $x$ and the value of $y$.
Method 1:
$$z + z_1 = 2z - z_2 ...
2
votes
1answer
27 views
Question on square roots of complex functions
LeI have a function $f(x)$ where $x\in R$ and $|x|<1$. Now, I can write $f(x)=\sqrt{1-x^2}g(x)$ for some non-divergent $g(x)$ in the domain of definition of $x$.
Can I write the following:
$$\sqrt{...
-1
votes
0answers
41 views
complex polynomial
I have a polynomial $P(z) = \lambda z - z^2$. I need to iteratively put $z = \lambda z - z^2$ in it. So, for the first two steps we have:
1) $P(z) = \lambda z - z^2$.
2) $P(z) = \lambda(\lambda z - ...
2
votes
1answer
57 views
Prove that $\frac{1}{2}+ \sum_{k=1}^n \cos (k\theta) = \sin((n+ \frac{1}{2})\theta)/2\sin\frac{\theta}{2}$ [duplicate]
Suppose $\sin \frac{\theta}{2} \neq 0$ . Prove that
$$\frac{1}{2}+ \sum_{k=1}^n \cos (k\theta) = \frac{\sin[(n+ \frac{1}{2})\theta]}{2\sin\frac{\theta}{2}}$$
The question also give the hint,
$$z=\...
-5
votes
1answer
50 views
What is the value of $({1+i})^i$ [on hold]
We know that $i=\sqrt{-1}$.
So what can be value of$({1+i})^i$.?
0
votes
1answer
76 views
Complex expression simplification
I am simplifying a complex expression like
$\mathcal{Im}[(Ae^{ip}+Be^{-ip})T^*e^{-2ip}]=|T|^2$sin $p$
to arrive at something like $\mathcal{Im}[A+B+Te^{2ip}]=0$,
My Try is the following, first I ...
0
votes
0answers
31 views
How to show that $\exists N \in \mathbb{N}$ s.t $|a_n|^{1/n} < \frac{1}{R} <\frac{1}{r}$, where $\limsup |a_n|^{1/n} = \frac{1}{R} $
In the book of Functions of One Complex Variables by Conway, at page 31, it is claimed that
However, I cannot understand the existence of such an $N$ that makes $|a_n|^{1/n} < 1/r$. I mean as far ...
2
votes
3answers
44 views
Write $-3i$ in polar coordinates.
can someone please help me with a trivial question?
Write $-3i$ in polar coordinates.
So $z=a+bi=rcis\theta$ with $r=\sqrt{a^{2}+b^{2}}$ and $\theta = arctan\frac{b}{a}$. However, what if $a=0$ ...
2
votes
2answers
43 views
Roots of unity and large expression
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \...
0
votes
1answer
31 views
Let $z=\operatorname{cis}\theta \in \mathbb{T}$ with $\theta \in \mathbb{Q}$. Show that the order of $z$ is infinite.
I am stuck on the following question:
Let $z=\operatorname{cis}\theta \in \mathbb{T}$ with $\theta \in \mathbb{Q}$. Show that the order of $z$ is infinite.
I am trying to understand Daniel Robert-...
0
votes
0answers
34 views
Maximum and minimum of complex functions [on hold]
Let $f(z) = z^7 + z^6 + z - 2018, $ where $z = a + bi.$ The function $g(z) \colon \mathbb{C} \to \mathbb{R}$ defined by $g(z) = |f(z)|$ admits a maximum $x$ and minimum $y$ in the compact $A_n = \{ z ...
0
votes
0answers
31 views
Can I write $(ix+\sqrt{1-x^2})=\sqrt{ix+\sqrt{1-x^2}}*\sqrt{ix+\sqrt{1-x^2}}$ when $x \in (-1,1)$?
Can I write $(ix+\sqrt{1-x^2})=\sqrt{ix+\sqrt{1-x^2}}*\sqrt{ix+\sqrt{1-x^2}}$ when $x \in (-1,1)$?
If not, then what is the rule for breaking up complex functions in square roots?
Sorry if this is ...
0
votes
0answers
15 views
With regards to finding a Fourier transform of a window function, how do I arrive at the sinc function?
I have a follow up to this question regarding the derivation of a sinc function from a complex exponential, because I haven't been able to figure out how the introduction of a scaling factor will ...
-2
votes
0answers
31 views
What is the definition of a complex equation? [on hold]
What is the definition of a complex equation?
0
votes
1answer
39 views
Is there an extension of the complex plane that is a group under addition and multiplication?
That is, having both $0$ and $1$ as the identity of $+$ and $\cdot$ respectively, and with an inverse for each ($-a$, $a^{-1}$) satisfying all the group laws.
EDIT:
Forming a group with $0$.
2
votes
0answers
30 views
The height of the Mandelbrot Set [duplicate]
A simple question, but one I am unable to find the answer to; What is
$$\sup\{y:x+y\cdot i \in M\}$$
Where $M$ is the Mandelbrot set. How is this constant calculated? Is it a known constant? Etc...
1
vote
1answer
72 views
Why the conditions $w(0)=0$ and $w(2)=\infty$ map the region $|z-1|<1$ onto the region $\Re w>0$?
I want to find a linear fractional transformation which maps the region $D$ of the $z$-plane onto the region $G$ of the $w$-plane, where
$D=\{z;|z-1|<1\},~G=\{w;\Re w>0\}$
This is an exercise ...
0
votes
2answers
64 views
Cube root of a complex number
I'm reading Linear Algebra Done Right by Sheldon Axler. In Chapter $1$, Exercise $A$, #2, it states:
Show that $(-1 + \sqrt{3i})/2$ is a cube root of $1$.
The solution on linearalgebras.com shows ...
2
votes
1answer
50 views
Can you convert a non “normal” complex square matrix into a “normal” one?
I've read the definition that "a complex square matrix ${\bf A}$ is normal if it commutes with its conjugate transpose".
I've also read that "A is a normal matrix iff there exists a unitary matrix U ...
0
votes
1answer
14 views
How the prove that the equality in the triangle inequality holds only if $z = tw$ or $w = cz$ for some $t,c \in \mathbb{R}^{nn}$
In the book of Functions of One Complex Variable by Conway, at page 3, it is stated that
[...] On encountering an inequality one should always ask for
necessary and sufficient conditions that ...
-1
votes
1answer
34 views
What does this set mean of $\mathbb{C} $ in complex plane? [on hold]
A = {$z \in \mathbb{C} ; Im(\dfrac{z-a}{b})>0, b \neq0$};
1
vote
4answers
67 views
Find the value of complex expression $\left(\frac{\sqrt{3}+i}{2}\right)^{69}$
Find the value of
$$\left(\dfrac{\sqrt{3}+i}{2}\right)^{69}.\DeclareMathOperator{\cis}{cis}$$
I tried to solve this complex expression by converting it into polar form.
I expressed it in polar ...
5
votes
1answer
56 views
Geometric intuition for the complex shoelace formula
The complex shoelace formula for the signed area of a triangle with vertices given by the complex numbers $a, b, c$ is $$\frac{i}{4}
\begin{vmatrix}
1 & 1 & 1 \\
a & b & c \...
0
votes
1answer
54 views
Deriving the Cubic Formula
I’m going through the derivation for the Cubic Formula.
Question: I know I am missing something painfully obvious, but how does $v=-\frac{h}{u}$ give the last two roots in $(5)$?
Quickly passing ...
-3
votes
0answers
35 views
Problem on complex numbers regarding double summation [on hold]
How does $$\sum_{j,k=1}^n a_je^{ix_j}a_ke^{-ix_k}$$ equal
$$\left|{\sum_{j=1}^na_je^{ix_j}}\right|^2$$
-1
votes
2answers
25 views
Finding the number of points on unit circle satisfying a criteria. [on hold]
Find the number of numbers$(z)$(Or the number of solution for $z$) on the unit circle such that :-
$z^{6!}-z^{5!}$ is a real number.
0
votes
1answer
28 views
Derivate of vector : transpose, conjugate and conjugate transpose
Let $x$ and $y\in \mathbb{C}^{K\times 1}$ and $H\in \mathbb{C}^{K\times K}$ a diagonal matrix.
$\bar{x}$ denotes the complex conjugated, $x^{T}$ denotes the transpose and $x^{*}$ denotes the complex ...
1
vote
1answer
28 views
What are all the angle-preserving $\mathbb{R}$-linear maps $T:\mathbb{C}\rightarrow\mathbb{C}$?
I know that linear maps of the form $T(z)=\alpha z,z\in\mathbb{C}$ preserve angles between two complex numbers. However, I know that every $\mathbb{R}$-linear map can be uniquely written as $T(z)=\...
1
vote
5answers
44 views
When should I measure the angle of a complex number clockwise or anticlockwise?
In this diagram theta is measured anticlockwise.
How would I know from which side to measure the angle?
In this diagram theta is measured clockwise.
0
votes
4answers
56 views
Express $z = \dfrac{3i}{\sqrt{2-i}} +1$ in the form $a + bi$, where $a, b \in\Bbb R$. [on hold]
Express $$z = \frac{3i}{\sqrt{2-i}} +1$$ in the form $a + bi$, where $a, b \in\Bbb R$.
I figure for this one I multiply by the conjugate of $\sqrt{2 +1}$? But I’m still struggle to achieve the form $...
0
votes
3answers
44 views
Complex roots with two variables [on hold]
$a + ai$ is a root of $x^2 − 6x + c = 0$, where $a, c ∈ \mathbb{R}$. Find all possible roots and all possible values of $c$.
3
votes
1answer
53 views
Homomorphism from $\mathbb R^2\to \mathbb C$
Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows:
$(a,b)(c,d)=(ac,bd)$
1
vote
0answers
36 views
Pole in limit expression
I would like to compute
$$
f(E,q):=\lim_{\epsilon\to 0}\text{Im}\left[ \frac{(E^2-q^2)^2-i\epsilon}{(E^2-q^2)^4+\epsilon^2}\left( E^2 + \alpha(m+in) \right) \right]
$$
with an infinitesimal ...
0
votes
1answer
26 views
Finding real and imaginary parts of $\frac{1-e^{2i \pi x}}{R \left(1-e^{\frac {2i \pi x}{R}} \right)}$
I have a function given by
$$\frac{1-e^{2i \pi x}}{R \left(1-e^{\frac {2i \pi x}{R}} \right)}$$
Using Euler's formula, I expand into real and complex components:
$$\frac{1-\cos 2 \pi x-i\sin2 \pi x}...
0
votes
0answers
20 views
justification for the definition of complex magnitude
The other sets of numbers, natural, rational, integer, real all have theories that are motivated by the models that inspired them and by consistency. The addition of naturals is associative because it ...
0
votes
1answer
30 views
Symmetrical Components in time function equation representation
I have equation which I can not understand until end, so your help I would appreciate.
I start with equation No. 1 which is clear:
The relations between the phase quantities $ \underline G_r,\...
2
votes
1answer
49 views
Complex number raised to irrational power
Why are there infinitely many values if a complex number is raised to an irrational power? I have to prove this, and don't know how. I am thinking Cauchy sequences, but I don't know how to start. ...
2
votes
4answers
78 views
Compute $(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)$ where $\alpha$ is the complex 5th root of unity with the smallest positive principal argument
I just started on the topic Complex Numbers and there is a question that I am stuck on.
The question is:
If $\alpha$ is a complex 5th root of unity with the smallest positive principal argument, ...
2
votes
2answers
30 views
find order of pole
$f(z) = \frac{1}{2-e^z}$
I found that $f$ is holomorphic on $\mathbb{C}-A$, where $A$ is the discrete set $A = \{\log(2) + 2\pi ik, k \in \mathbb{Z} \}$. If I now take $z_k = \log(2) + 2\pi ik$, then ...
3
votes
2answers
49 views
Find the maximum of the $| \left( w + 2 \right) ^3 \left( w - 3 \right)^2|$ with $|w|=1$
Let $w \in \mathbb{C}$, and $\left | w \right | = 1$. Find the maximum of the function $| \left( w + 2 \right) ^3 \left( w - 3 \right)^2|$
Since $$|(w+2)^3(w-3)^2|=|w^5-15w^3-10w^2+60w+72|$$
Let $...
3
votes
3answers
82 views
Algebraic trick to map $|z|<2$
Suppose that we want to find the image of the region $|z|<1$ under the mapping $w=\frac z{z+1}$. Since $z=\frac{-w}{w-1}$ we should have $|\frac w{w-1}|=|\frac{u(u-1)+v^2-iv}{(u-1)^2+v^2}|<1$ or ...
8
votes
2answers
170 views
+250
Can anything be proven about this complex variant of the Collatz problem, or is it just as intractable?
Given a Gaussian integer $z = a + bi$, where $a, b \in \mathbb{Z}$, $i = \sqrt{-1}$, iterate the function $$f(z) = \frac{z}{1 + i}$$ if $z$ has even Gaussian norm (that is, both $a$ and $b$ are odd, ...
1
vote
1answer
36 views
Generate fractals in the complex plane
I was looking at this bunch of fractals from the Wikipedia page "Frattale" (just the italian for Fractal).
z = z^2 + c
z = z^3 +c
z = z^4 +c
So I thought "Hey, I should write a program that lets ...
-1
votes
0answers
40 views
How do you represent the $\sqrt{9i}$ in the form $a+bi$? [closed]
I'm a bit confused as to how I can write the $\sqrt{9i}$ into the form $a+bi$
Sqrt of 9i
2
votes
1answer
34 views
For what integers $n$ is $-1$ an $n$th root of unity?
Can someone please verify my answer? I feel like I made it too complicated or I am missing something.
For what integers $n$ is $-1$ an $n$th root of unity?
There are two values on the unit circle ...
2
votes
0answers
38 views
Application of complex numbers [closed]
I have read complex numbers recently.While going through my quora feed i found out their actual application that how they can be used to analyze circles, ellipse, parabola and other curves in ...
0
votes
3answers
36 views
Graphic representation of the complex eigenvector of a rotating matrix
The eigenvector of the matrix
$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$
is
$$\begin{bmatrix}1\\i\end{bmatrix}$$
with its eigenvalue $-i$
$$-i\begin{bmatrix}1\\i\end{bmatrix}=\begin{...
-2
votes
3answers
32 views
How to algebraically determine locus of points of $|z|=|z-2|$? [closed]
How would I algebraically determine the locus of points in the $z$-plane that satisfy the equation $|z|=|z-2|$?
