Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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
-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 ...