Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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3
votes
3answers
68 views

Are Imaginary Numbers as $Real$ as Real Numbers?

In my Abstract Algebra book, ``A First Course in Abstract Algebra,'' by Fraleigh, the author seems to suggest that imaginary numbers are as $real$ as the real numbers, by asserting, for example, that ...
2
votes
1answer
39 views

Consider $az^2+bz+c=0$ where $a,b,c$ are all complex numbers

Consider $az^2+bz+c=0$ where $a,b,c$ are all complex numbers. What is the condition (ie the relation between $a,b,c$) for which the given quadratic has both real roots? I took the conjugate of the ...
-4
votes
1answer
38 views

What is $5$ taken to the power iota? [closed]

Could someone please solve this. I tried to put it in Euler's form but that wasn't any use because $$5 = 5e^{i\cdot0}=5.$$
-1
votes
0answers
44 views

Complex Numbers problem from JEE examination

Question: If $\omega$ is the imaginary cube root of unity such that $$\left|\left(\sum_{r=1}^n\left(r\sum_{p=1}^r\left(\omega^{p-1}\right)\right)\right)-155\omega\right|=\left(\sum_{r=1}^n\left(r\sum_{...
1
vote
1answer
27 views

Appearance of divisor sum in a certain series

Let us consider the following equivalence between two series, where $z \in \mathbb{C}$. I want to show that this equivalence indeed holds. $$\sum_{n,d=1}^{\infty} d^{k-1}e^{2\pi i d n z} = \sum_{n=1}^\...
1
vote
1answer
24 views

Solve complex equation with $\overline z$ [duplicate]

I need help solving this task, if anyone had a similar problem, it would help me. The task is: Solve the equation in a set of complex numbers. $z^3=\overline z$ I tried this : $z^3=\overline z\\\frac{...
1
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0answers
27 views

a minimization problem about complex number, sum with odd number, when $|z_i=1|$, $s_i$∈ {+1,-1} and $w_i$=1

There are some complex number $z_1=1,θ=120°$,because $θ=120°$, we can know that $z_2=cos⁡(120°)+isin⁡(120°)= -1/2+i √3/2$, and $z_3=cos⁡(240°)+i sin⁡(240°)= -cos⁡(120°)-i sin⁡(120°)= -1/2-i √3/2$. (...
5
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2answers
77 views

Is there a reason why $\text{Arg}(z)$ behaves like a logarithm?

In school, we learnt how to prove some of the basic properties of $\text{Arg}$, one of them being $$ \text{Arg}(z_1)+\text{Arg}(z_2)=\text{Arg}(z_1z_2) $$ We did this by writing $z_1$ and $z_2$ in the ...
0
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1answer
30 views

Finding cube roots of a unity - proper explanation is needed

SQUARE ROOT Let's say that I have: \begin{align*} x^2 &= 9 \end{align*} And I $\sqrt{\phantom{x}}$ an entire equation to get: \begin{align*} x_{1,2} &= \sqrt{9}\\ x_{1} &= +3\\ x_{2} &=...
0
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0answers
12 views

Using relations between polynomials and their roots to compute sums of complex numbers

I'm reading an article about Geometric barycentres of invariant measures for circle maps. During a proof of a theorem the author shows an expression like this $$b(\mu)=\frac{e^{i\varphi}}{q} \left( \...
3
votes
1answer
68 views

logarithm of complex number

Generally for logarithms If I have $2^4=16$ then it means $\log_2(16)=4$ (Here 2 is the base) So the value of logarithm basically tells us how many times to multiply the base for the number. When we ...
-1
votes
1answer
32 views

Complex numbers - solve equation

I need help solving this task, if anyone had a similar problem, it would help me. The task is: Solve the equation where they are $z\in \mathbb{C},m\in \mathbb{N}, n\in \mathbb{N}$ $z^n=z^{-m}$ I tried ...
1
vote
4answers
74 views

If complex number $a, b, c, d,$ and $|a|=|b|=|c|=|d|=1$, why $|a(c+d)|+|b(c-d)|\leq 2\sqrt{2}$?

If we have 4 complex number $a, b, c, d,$ and $|a|=|b|=|c|=|d|=1$, So, how to prove that $|a(c+d)|+|b(c-d)|\leq 2\sqrt{2}$? I try to separate $|a(c+d)|+|b(c-d)|$ to $|a||(c+d)|+|b||(c-d)|$ than I get ...
1
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0answers
14 views

Confused by the Positivity property of an Inner Product when the field is $\mathbb C$ - Attempted Proof

In my book "Linear Algebra as an Introduction to Abstract Mathematics", the inner product on vector space $V$ over field $\mathbb F$ is defined as the following map: $\langle \cdot,\cdot \...
0
votes
2answers
45 views

Extraneous solutions to $i^{2/3}$

I want to find the value of $$i^{2/3}$$ Here was what I tried: $$i^{2/3} = (i^{2})^{1/3} = -1^{1/3} = (-1^{2})^{1/6} = 1$$ I know that I could have also stopped at the third step, since $$-1^{1/3} = -...
1
vote
1answer
41 views

Prove that $\zeta^{a} = \zeta^{b}$ if and only if $o \left( \zeta \right) \mid a-b$

Show that for all complex roots of unity $\zeta$, and for integers $a$ and $b$, $\zeta^{a} = \zeta^{b}$ if and only if $o \left( \zeta \right) \mid a-b$ and $1, \zeta, \zeta^{2}, ...,\zeta^{o\left(\...
1
vote
1answer
31 views

Converging complex sequence

I would like to prove the following fact : if $(a_n)$ is a complex sequence such that : $|a_n|=1$ for every $n\in \mathbb N$ $a_{n+1}-a_n \rightarrow 0$ $(a_n^p)$ converges (for a fixed integer $p\...
-1
votes
1answer
19 views

Are there any transforms, that flip r and phi in polar coordinates?

Are there any transform, that flips $r$ and $\phi$ in polar coordinates? I.e. for each complex number $z = r e^{i \phi}$ it gives $T(z) = \phi e^{i r}$ Of course, this transform should not be ...
0
votes
5answers
54 views

Solve for $z \in \mathbb{R}$ : $z^6 = -64$

I can only think of the solutions 2i and -2i, but there should be more solutions. I am very new to complex numbers and equations and was wondering if anyone could help with the following question: ...
0
votes
0answers
80 views

Solving $\alpha z + \beta \bar z + \gamma = 0$

I attempted to find a general solution for the presented equation : $$\alpha z +\beta \bar z + \gamma =0$$ with $z,\alpha,\beta,\gamma \in \mathbb C$, my demonstration goes as below:$$\alpha z + \beta ...
-1
votes
2answers
57 views

Complex numbers/Prove equality [closed]

I need help solving this task, if anyone had a similar problem, it would help me. The task is: If the numbers $x,y,\theta$ are real and if it is $\frac{5+4\cos(\theta)}{2+\cos(\theta)+i\sin(\theta)}$ ...
0
votes
3answers
51 views

How to find complex solutions of equations? [duplicate]

I need help solving this task, if anyone had a similar problem, it would help me. The task is: Determine all solutions of the equation in the set of complex numbers. $(1-z)^5=z^5$ I thought I could ...
1
vote
1answer
57 views

Find the sum of infinite series $\cos{\frac{\pi}{3}}+\frac{\cos{\frac{2\pi}{3}}}{2}+..$

Find the sum of infinite series $$\cos{\dfrac{\pi}{3}}+\dfrac{\cos{\dfrac{2\pi}{3}}}{2}+\dfrac{\cos{\dfrac{3\pi}{3}}}{3}+\dfrac{\cos{\dfrac{4\pi}{3}}}{4}+\dots$$ I tried to convert it to $\mathrm{cis}$...
1
vote
2answers
46 views

Solving $z^{n}Re(z)=z^{-n}Im(z)$

So, I came across this equation and started playing around with it to see if I could solve it exactly. The first thing I noticed is that if I assume $z$ lies in the real or the imaginary axis the only ...
0
votes
0answers
21 views

Verification of substitution of variable in a complex polynomial equation?

I am writing as follows: The third-order polynomial of a complex variable $x,m,a,b \in \Bbb C$ is written as, $m = ax + b{\left| x \right|^2}x$ Where, ${\left| x \right|^2} = x{x^*}$ represent norm 2 ...
-1
votes
3answers
30 views

Complex numbers proof using modulus and conjugates [duplicate]

The question: If $\alpha , \beta$ are complex numbers where $\alpha \ne \beta$ and $|\alpha|=1$. Prove that $|\frac{\alpha \overline{\beta}-1}{\alpha-\beta}| =1$
0
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0answers
24 views

Derivative of this complex function

Assume that I have the following complex function $$f(Z)=\frac{1}{2\pi i}\left(-0.5a(Z-1)\ln \left( \frac{Z-1}{Z+1} \right) +0.5b(Z+1)\ln \left( \frac{Z-1}{Z+1} \right) \right)$$ where $Z$ is some ...
11
votes
4answers
266 views

Area as a complex number?

So I was trying to solve the following integral - $$\int_{2006}^{2020} \frac{\ln^{i}(\psi^{i})}{\psi} d\psi$$ Wasn't that bad actually (you can find the question and it's solution here). But I wasn't ...
0
votes
0answers
21 views

Conversion of complex number/quaternion to real vector

Is there a notation/function for "converting" (for want of a better word) a complex number or quaternion into a real vector with a dimension of 2 or 4? E.g. $2 + 3i + 4j + 9k$ would be $(2, ...
7
votes
8answers
136 views

Solving $z^4=(2+3i)^4$

To solve the equation, I calculated right side: $z^4=(2+3i)^4=(-5+12i)^2=-119-120 i$ And then I get the correct answer: $z_k=\underbrace{\sqrt[8]{119^2+120^2}}_{\sqrt{13}} \times Cis(\cfrac{\pi+\tan^{-...
0
votes
2answers
33 views

Let $S$ be the set of all points $z$ in the complex plane such that ${\left(1+\frac{1}z \right) }^4=1$. Then, the points of $S$ are -

QUESTION: Let $S$ be the set of all points $z$ in the complex plane such that $${\left(1+\frac{1}z \right) }^4=1$$ Then, the points of $S$ are (choose the correct option) (A) vertices of a rectangle ...
0
votes
0answers
28 views

Need a proper reasoning for a proposed claim on complex numbers.

QUESTION: I have just come to know that, if it is given that $\frac{z-a}{z-b}$ is purely imaginary, where $z \in \Bbb{C}$, then it implies that the angle between the vectors $z-a$ and $z-b$ is $\...
0
votes
4answers
78 views

How do I find all complex numbers $z$ such that $z^2+|z|=0$? [duplicate]

I need help solving this task, if anyone had a similar problem, it would help me. The task is: Determine the complex numbers $z$ from the condition. $$z^2+|z|=0$$ By my logic, the solutions are : $-...
2
votes
2answers
43 views

Infinite limit of a complex function

Question $$ \lim_{x \to 0}\frac{1}{x^2 + i x} $$ The related info (following solution and plot taken from Wolfram Alpha) $$ \lim_{x \to 0^-}\frac{1}{x^2 + i x} = i \infty $$ $$ \lim_{x \to 0^+}\frac{1}...
0
votes
3answers
41 views

Problem related to the construction of a heptadecagon.

If we have a list of the $17$ roots of unity of a 17 sided polygon such as here: $A=\zeta +\zeta^{9} +\zeta^{13} + \zeta^{15} + \zeta^{16} + \zeta^{8} + \zeta^{4} + \zeta^{2} $ $B=\zeta^{3} +\zeta^{10}...
2
votes
4answers
51 views

Region $S$ represented by $a+b\omega +c{ \omega }^{ 2 }$

Let $\omega =-\dfrac { 1 }{ 2 } +\dfrac { \sqrt { 3 } }{ 2 } i$ and $S$ denote the set of all the complex numbers in the Argand plane of form $a+b\omega +c{ \omega }^{ 2 }$ , where $a,b\in [0,1]$ ...
1
vote
1answer
18 views

How to find partial fraction of a complex expression

I am learning complex analysis, and i am asked to find the series for $$f(z)=\frac{1}{(z-1)(z-2i)}$$ , it says that it can be done at the point z=1+i . It then follows in the book to write $$\frac{1}{...
8
votes
1answer
130 views

If the complex sequence $u_{n+1}=f(u_n)$ has only one limit point, then it converges

I would like to prove that if $f:\mathbb{C}\to\mathbb{C}$ is continuous and if a sequence $u$ defined by : $\forall n\in\mathbb{N},\,u_{n+1}=f(u_n)$ has only one limit point (not sure of the ...
1
vote
1answer
24 views

Sum of binomial numbers alternating the sign with complexes.

Prove ${n \choose 1}- \frac{1}{3}{n \choose 3}+ \frac{1}{9}{n \choose 5}-\frac{1}{27}{n \choose7}+...= \frac{2^n}{3^{\frac{n-1}{2}}}\sin({\frac{n\pi}{6}})$ I used the binomial expansion $(1+x)^n$ ...
1
vote
0answers
37 views
+50

How may one obtain two real-valued, closed-form functions of x and y of a complex variable or function z?

I am by no means a mathematician, but a programmer, and somewhat related and by no means a trivial task, I am looking for real-valued, closed-form, arbitrary precision approximations of the circular ...
1
vote
2answers
54 views

Let $|z_1|=|z_2|=|z_3|=3$. Then find minimum value of $|z_1+z_2|^2+|z_2+z_3|^2+|z_3+z_1|^2$ is?

$$(z_1+z_2)(\bar z_1 +\bar z_2)+(z_2+z_3)(\bar z_2+\bar z_3)+(z_3+z_1)(\bar z_3+\bar z_1)$$ $$=2(|z_1|^2+|z_2|^2+|z_3|^3) + Re (z1\bar z_2) + Re(z_2\bar z_3)+Re(z_3+\bar z_1)$$ Assuming $z=x+iy$ $$=54 ...
1
vote
1answer
71 views

Let $z=x+iy$, where $x,y\in \mathbb{Z}$. Find the area of the octagon whose vertices are roots of $(z\bar z)|z^2-\bar z^2|=1200$

$$|z^2||z-\bar z||z+\bar z|=1200$$ $$(x^2+y^2)(|2x|)(|2y|)=1200$$ $$(x^2+y^2)(|xy|)=300$$ I don’t think there is any realistic way to obtain $x$ and $y$ other than hit and trial. I am asking this ...
-6
votes
0answers
29 views

complex numbers , college mathematics [closed]

Find all the values of $log(1-i)$ and $\log(3-2i)$ Find all values for the following expressions: $5^i$ and $\log(1+i)^{πi}$
1
vote
1answer
17 views

Domain of $\operatorname{Arg}(1/z)$: $\operatorname{Re}(z) \neq 0$

The book I'm using is Complex Variables and Applications, 9th Ed by Brown and Churchill. I'm confused about exercise 14.1.b (pg 43): For each of the functions below, describe the domain of definition ...
0
votes
3answers
35 views

Finding the rank of a matrix $A$

Find the rank of the $n \times n$ matrix $A = [i + j]_{i,j \le n}$ (over $C$). C here should be the complex space; although i am having trouble interpreting what A exactly is, I do not understand the ...
0
votes
0answers
21 views

Complex Matrix Multiplication and Orthogonality

I am dealing with an electromagnetic problem which relates the electric field to the magnetic field in the frequency domain via the wave impedance tensor. In a 1-D medium, the equation is: $$\mathbf{E}...
0
votes
0answers
20 views

Cauchy's integral theorem proof from Green's theorem

I am studying Cauchy's integral theorem from shaum's outline,the theorem states that Let $f(z)$ be analytic in a region $R$ and on its boundary $C$. Then $$\oint_{C}f(z)dz=0$$ After the statement of ...
0
votes
0answers
31 views

Can $\operatorname{Re}(a+bi)^n$ be overlapped with $a,b\in\mathbb{Q}$ fixed? (Just changed the $\mathbb Z$ in my earlier question to $\mathbb Q$)

This is a generalization of my earlier question, Can $\operatorname{Re}(a+bi)^{n}$ be overlapped with $a,b\in\mathbb{Z}$ fixed?, just changing $\mathbb Z$ into $\mathbb Q$. Is there any non-trivial ...
2
votes
2answers
30 views

Let $s,t,r$ be non zero complex numbers and $L$ is is set of solutions of $z=x+iy$ of the equation $sz + t\bar z+r=0$

Prove that $L$ is a singleton set if $|s|\not =|t|$ And, prove that $z$ is a straight line if $L$ is not singleton Solving the equation, I got $z=\frac{\bar s r -\bar r t}{|t|^2-|s|^2}$ I personally ...
1
vote
3answers
44 views

finding the principal arguement

I have $(1+i)^{13}$ and I need to find the principal argument. I did this: $(1+i)^{13}$ = $(2)^{13/2}(cos(\frac{13pi}{4}) + isin(\frac{13pi}{4}))$ using De Moivre's Theorem, but I dont know where to ...

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