Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
15,210
questions
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
vote
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
votes
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
votes
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
votes
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
$$b(\mu)=\frac{e^{i\varphi}}{q} \left( \...
3
votes
1answer
68 views
logarithm of complex number
Generally for logarithms
If I have $2^4=16$ then it means $\log_2(16)=4$ (Here 2 is the base)
So the value of logarithm basically tells us how many times to multiply the base for the number.
When we ...
-1
votes
1answer
32 views
Complex numbers - solve equation
I need help solving this task, if anyone had a similar problem, it would help me.
The task is: Solve the equation where they are $z\in \mathbb{C},m\in \mathbb{N}, n\in \mathbb{N}$
$z^n=z^{-m}$
I tried ...
1
vote
4answers
74 views
If complex number $a, b, c, d,$ and $|a|=|b|=|c|=|d|=1$, why $|a(c+d)|+|b(c-d)|\leq 2\sqrt{2}$?
If we have 4 complex number $a, b, c, d,$ and $|a|=|b|=|c|=|d|=1$, So, how to prove that $|a(c+d)|+|b(c-d)|\leq 2\sqrt{2}$？
I try to separate $|a(c+d)|+|b(c-d)|$ to $|a||(c+d)|+|b||(c-d)|$ than I get ...
1
vote
0answers
14 views
Confused by the Positivity property of an Inner Product when the field is $\mathbb C$ - Attempted Proof
In my book "Linear Algebra as an Introduction to Abstract Mathematics", the inner product on vector space $V$ over field $\mathbb F$ is defined as the following map:
$\langle \cdot,\cdot \...
0
votes
2answers
45 views
Extraneous solutions to $i^{2/3}$
I want to find the value of
$$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$$
I tried to convert it to $\mathrm{cis}$...
1
vote
2answers
46 views
Solving $z^{n}Re(z)=z^{-n}Im(z)$
So, I came across this equation and started playing around with it to see if I could solve it exactly. The first thing I noticed is that if I assume $z$ lies in the real or the imaginary axis the only ...
0
votes
0answers
21 views
Verification of substitution of variable in a complex polynomial equation?
I am writing as follows:
The third-order polynomial of a complex variable $x,m,a,b \in \Bbb C$ is written as,
$m = ax + b{\left| x \right|^2}x$
Where, ${\left| x \right|^2} = x{x^*}$ represent norm 2 ...
-1
votes
3answers
30 views
Complex numbers proof using modulus and conjugates [duplicate]
The question:
If $\alpha , \beta$ are complex numbers where $\alpha \ne \beta$ and $|\alpha|=1$.
Prove that $|\frac{\alpha \overline{\beta}-1}{\alpha-\beta}| =1$
0
votes
0answers
24 views
Derivative of this complex function
Assume that I have the following complex function
$$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 ...
