Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
0
votes
1answer
29 views
For $z \in \mathbb{C}$, the only point where the derivative for $f(z)=|z|^{2}$ exist is the origin.
Let $z=x+iy$, as $f(z)=z\bar{z}$ we have $f(z)=x^2+y^2$. Then we can define:
$$f(x,y)=u(x,y)+i v(x,y).$$
Where $u(x,y)=x^2$ and $v(x,y)=y^2$. To see $f$ has derivative in the origin I can see the ...
-1
votes
2answers
24 views
Complex numbers involving triangle ABC.
Please help me solve this problem, I've tried sketching a triangle on axes of Real against Imaginary number. But don't know how to proceed. Should I treat $v$ and $u$ as vectors? Should I apply $AB= v ...
1
vote
2answers
23 views
Isomorphism between C and another ring
Let the operations of addition and multiplication on the set
$K = {at+bu : a,b ∈ R}$,
where $t$ and $u$ are formal symbols, be defined as follows:
$(at+bu)+(ct+du) = (a+c)t+(b+d)u$,
$(at+bu)·(ct+du) ...
4
votes
2answers
34 views
How is $\sum_{k=0}^{5} {}^5C_k\sin(kx)\cos(5-k)x=16\sin(5x)$?
The original question was if $\sum_{k=0}^{5} {}^5C_k\sin(kx)\cos(5-k)x=N\sin(5x)$ then what is the value of N?
I plugged $\frac{\pi}{2}$ in place of x and N came out to be 16. But is there any ...
-1
votes
3answers
50 views
Prove: If $z\in \mathbb{C}$ and $|z|=1$, then $|z+1|^2+|z-1|^2=4$.
If $z\in \mathbb{C}$ and $|z|=1$, then $|z+1|^2+|z-1|^2=4$.
My attempt:
$|z|=1 \Leftrightarrow z=1^2=1$
Hence $|1+1|^2+|1-1|^2=4$
Is that enough?
1
vote
2answers
15 views
Computation of time average of a product of two oscillatory quantities
Can somebody please help me to prove the formula below
$\frac{1}{T} \int_{0}^{T} \operatorname{Re}\left[A e^{i \omega t}\right] \operatorname{Re}\left[B e^{i \omega t}\right] d t=\frac{1}{2} \...
-4
votes
0answers
27 views
what is the XOR output of the 16 bit input sequence 0001_1111_1100_1001? [on hold]
what is the XOR output of the 16 bit input sequence 0001_1111_1100_1001 ?
-1
votes
2answers
24 views
Infinite abelian subgroup of $SL(2,\mathbb{C})$ [on hold]
Are there any infinite abelian subgroups of the special linear group $SL(2,\mathbb{C})$?
1
vote
1answer
37 views
What is the value of $e^{3i \pi /2}$? [duplicate]
When solving for the value, we know that $e^{\pi i}=-1$ .
I am confused as to what is the right answer when you evaluate this.I am getting two possible answers: $e^{3\pi i/2}$ = $(e^{\pi i})^{3/2}$ so ...
2
votes
0answers
47 views
Completing the square with complex numbers
Dear MSE-community!
I have begun on a journey through Gamelin's Complex Analysis. In the first chapter is an exercise described in the Math.SE question "Show that the set $z$ satisfying $|z−z_0|=\rho|...
1
vote
1answer
34 views
sums of series involving complex numbers
I dont know how to use latex any good rescources to quickly learn from scratch, hence why I have photographed the question.
The question is as follows
question
question final part of working
The ...
1
vote
1answer
42 views
Determine a Constant
Let $\lambda \in C$ and that $\lambda \in B(\mu,r)$, a closed ball centered at $\mu \in C$, with radius $r < \mu$. I am trying to determine the value of a constant $\tau$ to guarantee that:
$$
|1-\...
0
votes
2answers
27 views
If $V=\overline V$ where $V$ is a vector space, then there exists a real basis of $V$ [on hold]
Considering $V$ to be a vector space over $\mathbb C$ then if $V=\overline V$ how can i assure there is a real basis?
My first consideration was that if $w \in V$ then $\bar w \in V$ so $Re(w) \in V$ ...
0
votes
2answers
53 views
Solve in $C$ : $p(x)=x^4+2x^3-x^2-2x+7=0$ [duplicate]
Find all root :
$p(x)=x^4+2x^3-x^2-2x+7=0$
Where $p(\alpha)=0$ ,
$\alpha=\sqrt{2}+\omega$
$\omega=e^{\frac{2iπ}{3}}$
My try :
Since : $\alpha$ root of equation then
$\bar\alpha$
also root ...
0
votes
1answer
36 views
What online graphing tools handle complex numbers well?
What online graphing tools handle complex numbers well?
Desmos is generally excellent by breaking functions down into their real and imaginary parts and plotting on the Euclidean plane. For example ...
0
votes
2answers
32 views
Can this exponential be complex valued?
My complex analysis is very sketchy, and I am a little stumped by the following - although it seems incredibly innocuous.
For $t\in\mathbb R$ and a fixed parameter $\alpha\in\mathbb R/\{0\}$ does it ...
0
votes
0answers
39 views
How is Im$[(z_1+z_2)(z_2+z_3)(z_3+z_1)]/(z_1 z_2 z_3) = 0$
How is Im$\frac{[(z_1+z_2)(z_2+z_3)(z_3+z_1)]}{(z_1 z_2 z_3)} = 0$ if $|z_1|=|z_2|=|z_3|=1$? My book says it is because $\frac{[(z_1+z_2)(z_2+z_3)(z_3+z_1)]}{(z_1 z_2 z_3)}$ is equal to its conjugate, ...
0
votes
2answers
32 views
Convergence of $\sum_{n=1}^\infty \frac{z^{n!}}{n^2}$ where $\vert z\vert = 1$?
We were asked to study the convergence of this series $$\sum_{n=1}^\infty \frac{z^{n!}}{n^2}$$
on the boundary of its convergence disc i.e.: $\vert z\vert = 1$
And a friend argumented something like ...
4
votes
2answers
62 views
How to show that $f(z)= 1/(z-a)$ is periodic with period $2\pi i$?
From Ahlfor's book, Complex Analysis, page 149:
"The particular function $1/(z-a_j)$ has the period $2\pi i$."
How can one see this is true?
I tried writing $z$ as $|z| e^{i \arg{z}}$ and using the ...
-1
votes
0answers
33 views
Find the number of subsets such that sum of the elements in each subset is divisible by $9$ [duplicate]
Given $A=\left\{1,2,3,\cdots 2019\right\}$, Find the number of subsets of $A$ such that sum of the elements in each subset is divisible by $9$
My try:
I was trying a very long way but a basic ...
6
votes
1answer
50 views
Powers of complex numbers.
It is known that if $\alpha$ is a complex number, then for example, the equation $x^2 = \alpha$ has $2$ solutions. In general, there are $n$ values for the $n$-th roots of a number. In other words, if ...
0
votes
4answers
87 views
How do I solve $z^6+z^5+z^4+z^3+z^2+z=0$? [on hold]
I tried to use de Moivre's formula to solve it, but I had no success.
Can you give me any tips?
0
votes
1answer
28 views
solving equations with complex numbers, with one and multiple equations
The question is as follows, I am being asked a question in which I have to solve for $a \in \mathbb{C}$. Find all $a\in \mathbb{C}$ such that
$a^3+1=0$, $a^6-1=0$ and $a^4-1=0$
3
votes
2answers
130 views
Are complex numbers two dimensional or one dimensional?
Complex numbers are represented as:
z = x + yi
This gives the impression that complex numbers are a real component plus an imaginary component. However, when doing math with complex numbers, they ...
0
votes
3answers
29 views
Complex number, finding complex plane
I have to find the complex plane region that is determined by the following condition:
$$|2z+3|\gt 4$$
I develop using my definition of $z$ as $a+bi$
and I get to
$$a^2+3a-b^2 \gt \frac{7}{4}$$
...
3
votes
1answer
47 views
Complex numbers problem with absolute value property
If a,b are complex numbers, k its an integer, $k \neq 0$ and $|a+k| + |b-k| + |a+b-k|=1$ then proof that $a,b$ are real numbers
I've tried
$a+k=x$ and $b-k=y$
Then I used absolute value ...
1
vote
4answers
39 views
$S=\left \{z\in\mathbb{C}| (z+i)^{n}=(z-i)^{n} \right \}$ $S=?$
If $n\in\mathbb{N}, n\geq2$ and $S=\big \{z\in\mathbb{C}| (z+i)^{n}=(z-i)^{n} \big \}$ then $S=?$ The right answer is
$$
S=\left \{\operatorname{ctg}\frac{k\pi}{n} |1\leq k\leq n-1;k\in\mathbb{N}\...
0
votes
3answers
81 views
Square root of a negative number
Correct me if I am wrong, $\sqrt{-4}=2i$.
But how do you explain it to a student?
We know $\sqrt{-1}=i$, but one cannot say $\sqrt{-4}=\sqrt{-1}\sqrt{4}=2i$ as the laws of indices can only be ...
1
vote
0answers
38 views
Using mathematical induction to prove products of trigonometric function
So I have this identity or result from studying polygons and complex numbers which states:
$$\prod_{j=0}^{n-1}\sqrt{\bigg(\cos\frac{\theta+2j\pi}{n}-\cos\frac{\theta}{n}\bigg)^2 +\bigg(\sin\frac{\...
1
vote
2answers
34 views
Simple modulus inequality
I want to determine all $z\in\mathbb C$ for which
\begin{equation}\tag{1}\label{1}
\left|\frac{2+z}{2-z}\right| \le 1.
\end{equation}
In fact, I already know that \eqref{1} is equivalent to $\Re(z)\...
1
vote
1answer
73 views
How to find the conjugate of $\sum_{n=0}^{m}n \cos(2\pi x n)$?
Supposing I want to find the conjugate of
$$\tag{1}\label{eq1}\sum_{n=0}^{m}n \cos(2\pi x n)$$
If I view (1) as a Fourier series then what would be the conjugate Fourier series and how would I go ...
0
votes
0answers
21 views
A problem about Argand diagrams
When plotting an Argand diagram, do I simply plot the points or do I need to draw arrows from the origin to the points?
Motivation: I self studied for my FP1 exams, which will be held today and ...
0
votes
3answers
35 views
How to find all solutions of $\cos(z) = 0$, where $z\in\mathbb{C}$? [duplicate]
I am stuck on finding all solutions of the equation $\cos(z) = 0$, where $z\in\mathbb{C}$.
I found this proof, however, I cannot figure out the logic behind the last few steps.
\begin{align}
\cos(z) &...
0
votes
1answer
48 views
How to Prove $(-1)^{\frac{1}{2}}$ = i? [duplicate]
What would be a formal way to prove the above statement?
0
votes
0answers
10 views
Computing the marginal of a complex Gaussian distribution
I have the following complex Gaussian conditional distribution
\begin{align}
p(x | \mu, \phi, \sigma) = \frac{1}{\pi^{\frac{1}{2}} \cdot \sigma } \exp \bigg( -\frac{1}{\sigma^{2}} \Big( \overline{x} -...
0
votes
2answers
19 views
Exponential complex numbers question [closed]
Find all solutions of $\mathbf{e^{4z}= −3 − 3i}$
I'm having a bit of trouble with the question above. I started out with $4z= \ln(-3-3i)$ but I don't know where to go from here.
Help would be much ...
0
votes
1answer
14 views
Conjugate of complex Harmonic function?
$$
\begin{array}{l}{\text { Let } f=u(x, y)+i v(x y), \text { where } u=\sin (x) \sinh (y)} \\ {\text { find the conjugate of } u(x, y) \text { if it is harmonic. }}\end{array}
$$
I am pretty lost on ...
0
votes
1answer
30 views
Complex number problem.
For any complex number z, prove that $|Re(z)| + |Im(z)| ≤ (√2)(|z|)$
In my approach, I assumed $z$ to be equal to $x + i y$
which leads me to
LHS = $√x² + √y²$
= $x + y$
And
RHS = $√2 (√...
0
votes
0answers
49 views
How to use complex number method to prove
that a quadrilateral whose two pairs of opposite sides are equal in length is a parallelogram. Please do not use any axioms of synthetic geometry but can use vector geometry.
Suppose four distinct ...
0
votes
1answer
21 views
Phasor/Harmonic Addition Formula/Theorem: Why can we take out the frequency out of an complex argument?
Harmonic Addition Theorem
Harmonic Addition Formula
Phasor Addition Theorem
Phasor Addition Formula
Those four name can be used as a keyword on google.
I haven't known the official name and think ...
9
votes
2answers
2k views
Is taking modulus on both sides of an equation valid?
This might look like a copy of another question, but what I'm about to propose here is new. There's this question,
Find the least positive integral value of n for which $(\frac{1+i}{1-i})^n = 1$
...
1
vote
1answer
64 views
What is the value of $|\sin(\cos\theta + i \sin\theta)| $ in complex analysis? [closed]
How would I compute the value to/simplify the following expression?
$$\left|\sin\left( \cos\theta + i \sin\theta \vphantom{M^M} \right) \right| $$
Can I use the fact that $\cos\theta + i \sin\...
0
votes
2answers
50 views
Is the $\pm$ sign used when finding the root of a negative number?
If $\sqrt{64}$ is equal to $\pm{}8$, is $-64$ equal to $\pm{}8i$, or just $8i$?
0
votes
0answers
22 views
Notation for scalar coefficients of a vector
I want to know if there is some existing notation for the "magnitude" of a vector when it is complex. For example, suppose I have the following vector:
$$\mathbb{v}=(2+i) \ \hat{x}$$
Is there any ...
0
votes
0answers
22 views
Proof of n radicals to a trigonometric expression
For all natural numbers n, let
A(n) = (2-(2+(2+.... +(2)^(1/2))^(1/2))^(1/2)) ^(1/2))
( n many radicals) then show that for n>=2
A(n) = 2 sin( π/2^(n+1))
I tried squaring the expression to ...
0
votes
0answers
10 views
Intuition behind the definition of rational convexity
For a compact set $K\subseteq\mathbb{C}^n$ Polynomially convex hull is defined to be the set $\{z\in \mathbb{C}^n: |p(z)|\leq ||p||_K\ \mbox{for all polynomials } p\}.$ On similar lines, for a ...
1
vote
2answers
52 views
Uniqueness Theorem Complex Analysis
I am having some trouble solving the following question for my practical final.
Does there exist an entire function f : C → C s.t. $f(1/n) = \frac{n^2}{n^2 + 1 }$ for all n in N
I don't think ...
2
votes
1answer
70 views
Using De Moivre's law to compute $(-\sqrt3+i)^{2/3}$
Question:
If $z=-\sqrt{3}+i$, then $z^{2 / 3} = ?$
My work (which is wrong but I am not sure why):
We can write $z = r(\cos\theta + i\sin\theta)$
$$r = \sqrt{(-\sqrt{3})^2 + 1^2} = \sqrt{4} = 2$...
0
votes
1answer
24 views
Inequality within complex
I step with a inequality and would like to know if it is truth...
$||(a_1-a_2)^2+i(b_1-b_2)^2||\leq ||a_1^2+ib_1^2||+||a_2^2+ib_2^2||,\quad \forall a_1,a_2\in\mathbb{R}$.
I tried to prove it but ...
0
votes
0answers
21 views
Find a basis for a set involving complex conjugates
How can I find a basis for this vector space over ℝ:
A={(u, v, v̄, -u) : u,v ∈ 𝐂}
(so u and v are complex numbers of the form a + bi and v̄ is the complex conjugate of v).
It's just the complex ...
