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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2
votes
2answers
45 views

Complex exponential to real

I'm not yet very good at complex number, so I would appreciate the following insight: How exactly do we arrive from $e^{\pi(1-i)}-e^{-\pi(1-i)}$ to $e^{-π}-e^π$, and why does ...
4
votes
2answers
52 views

Why are values greater than $\pi$ radians given as negative in exponential form?

Find the fifth roots of $-3+3i$ in exponential form. My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$ Wolfram ...
1
vote
1answer
29 views

Multiplying square roots of negative numbers

I am just learning more about complex numbers and a question popped up I can't figure out on my own, so I've posted it here. I already know $i^2=-1$ and $i=\sqrt{-1}$ (isn't it even true that $\pm ...
7
votes
1answer
463 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
3
votes
1answer
86 views

Triangle inequality- complex

I am trying to prove the triangle inequality purely algebraically. Let $z=x+iy$, $w=u+iv$. Then, $|z+w|^2$=$|(x+u)+i(y+v)|^2$=$(x+u)^2+(y+v)^2$=$x^2+2xu+u^2+y^2+2yv+v^2$ I tried the other way: ...
9
votes
9answers
193 views

How to solve $z^3 + \overline z = 0$ [duplicate]

I need to solve this: $$z^3 + \overline z = 0$$ how should I manage the 0? I know that a complex number is in this form: ...
0
votes
0answers
39 views

matching the powers of the coefficients of polynomials

Hi: The result of the following question is stated (as an "it is straightforward to show that" type of result) in an econometrics paper, the link of which I can provide. But I translated into a ...
9
votes
1answer
159 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
2
votes
2answers
54 views

Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side ...
56
votes
25answers
7k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
1
vote
4answers
61 views

The real part of a complex number such that $|z|=\max\{|z-2|,|z+2|\}$

If |z|=max{|z-2|,|z+2|} then - INFERRENCE - |Re(z)|=1 Is the inferrence incorrect? My approach is - |z|=|z-2| when |z-2| {i.e. distance of z from 2 is greater}is greater OR |z+2| when |z+2| is ...
2
votes
5answers
238 views

Definitions for complex numbers

I could not find this question anywhere else. But why are addition, subtraction, division, and other operations defines they are in complex numbers? Could they defined as something else?
4
votes
3answers
89 views

The imaginary unit, $i,$ and an alternate representation.

Recently, I began working with both complex, and imaginary numbers, and I looked at the complex number $i^{n}.$ If $n = 0, i^{n} = 1,$ $n = 1, i^{n} = i = \sqrt{-1},$ $n = 2, i^{n} = i^{2} = i ...
0
votes
2answers
42 views

Identity with complex numbers related to the Cauchy-Schwarz inequality

I have this equation $ a_j,b_j\in \mathbb{C} , j=1,2,...,n$ $$ \left| \sum\limits_{j=1}^n a_jb_j \right|^2 = \sum\limits_{j=1}^n |a_j|^2 \sum\limits_{j=1}^n |b_j|^2 -\sum_{1\leq i \leq j \leq n} ...
1
vote
6answers
78 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
1
vote
2answers
28 views

Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
0
votes
4answers
76 views

Sum of roots of unity, proving that $1+w+w^2…+w^{n-1}=0$ [closed]

If $w$ is a unit square of rank $n$ (meaning $w^n=1$), s.t $w$ is not $1$. Prove that $1+w+w^2.....+w^{n-1}=0$. We're pretty sure that we need to use induction, its easy to prove for $n=2$ but ...
2
votes
2answers
32 views

Fixed points of $\frac{1\pm \sqrt{1-|a|^2}}{\bar a}.$

Prove that $\phi_a(z)=\frac{a-z}{1-\bar az}$ , $0<|a|<1$ has exactly two fixed points ; one inside the unit disc and the other outside the unit disc. Putting $\phi_a(z)=z$ I find that there ...
4
votes
0answers
60 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
1
vote
2answers
75 views

Question about a step in the proof of the Cauchy-Schwarz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwarz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le \sum_{j=1}^n\vert ...
0
votes
4answers
57 views

How find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$?

Let $k > 1$ be a real number. How may one find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$? ...
1
vote
0answers
33 views

Complex numbers $x$, $y$, and $z$ are collinear, show that there exist $a,b,c$ for which $ax+by+cz=0$ where $a+b+c=0$

Suppose $x$, $y$, and $z$ are collinear (complex numbers). Prove that there exist real constants $a$, $b$, $c$, not all zero, such that $ax+by+cz=0$ where $a+b+c=0$. This is how I did it: First, ...
4
votes
2answers
93 views

What is the solution(s) to $1^i$?

$1^x$ is always $1$ with real numbers, but everything gets more complicated with complex numbers. Using Eulers formula, you know that $$e^{ix}=\cos(x)+i\sin(x)$$ If you make x=2π into this you'd get ...
0
votes
2answers
441 views

Find a solution to any single-variable equation

I know it is not possible to solve any equation of fifth degree and higher "using only a finite combination of the arithmetic operations and radicals in terms of the coefficients" (see on Wikipedia). ...
7
votes
8answers
897 views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
-1
votes
3answers
73 views

$ x+y = 1 $ and $ \frac{1}{x} + \frac{1}{y} = 1 $ Solve $ x^3 + y^3 $ [closed]

$x$, $y$ are complex numbers, $x$ and $y$ aren't $0$. $$ x + y = 1 $$ $$ \frac{1}{x} + \frac{1}{y} = 1 $$ $$ x^3 + y^3 = ? $$ Thank You!
3
votes
1answer
101 views

Finding an analytic function satisfying given two conditions.

Does there exists an analytic function $f:D\to D$ such that $f(1/2)=1/2$ and $f'(1/2)=-1$ ? If exists then find such a function. where , $D=\{z\in \mathbb C:|z|<1\}.$ I found that such a ...
1
vote
1answer
26 views

Checking whether points form a polygon in complex plane

If z^8=(z-1)^8 then the roots are 1) concyclic 2) form a polygonal 3)none I found the roots to be 1+cot(k.pi/8) for k is a natural number and less than 8. Then couldn't figure it out.
2
votes
3answers
65 views

Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

Let , $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$ denotes the semicircle traversed clockwise. Show that $$\lim_{r\to 0}\int_{C_r}\frac{e^{iz}}{z(z^2+1)}\,dz=-\pi i$$ I can not use the Jordan's ...
12
votes
2answers
175 views

Solving $z^z=z$ in Complex Numbers

I wanted to find all complex numbers $z\neq0$ such that $z^z=z$. I observed that $z=\pm1$ satisfies the equation. But I had problems when tried to find all the possible solutions since $z^z$ may take ...
0
votes
1answer
17 views

Complex inequality question

I am trying to understand why the following holds: \begin{align*} \Re((1-\imath)(A+B)) \geq \Re((1-\imath)A) - \sqrt{2}|B|, \end{align*} where, \begin{align*} A:= \sum_{x=1}^{[\sqrt{k}]} ...
2
votes
3answers
87 views

Imaginary numbers and polynomials question

I have a task which I do not understand: Consider $w = \frac{z}{z^2+1}$ where $z = x + iy$, $y \not= 0$ and $z^2 + 1 \not= 0$. Given that Im $w = 0$, show that $| z | = 1$. Partial solution (thanks ...
0
votes
1answer
51 views

Questions on whether imaginary number is larger than $0$

The imaginary number implies $i=\sqrt{-1}$ But this this not say whether $i$ is larger or smaller or equal to $0$ So i wonder if $i$ can be larger or smaller than $0$ and if so how do we see it
5
votes
1answer
97 views

Why is $\sqrt{xy}=\sqrt{x}\sqrt{y}$ also true when $x=-1$, making it $i$?

A : If: $$\sqrt{xy}=\sqrt{x}\sqrt{y}$$ only when $x,y>0$, B : Then why can I do this: $$\sqrt{-4}=\sqrt{4\times-1}=\sqrt{4}\sqrt{-1}=2i$$ which violates A since $y<0$ C : But why can I not ...
4
votes
2answers
293 views

Prove that: $\sin{\frac{\pi}{n}} \sin{\frac{2\pi}{n}} …\sin{\frac{(n-1)\pi}{n}} =\frac{n}{2^{n-1}}$

Using that: $$ x^{n - 1} + x^{n - 2} + \cdots + x + 1 = \left(x - w\right)\left(x - w^{2}\right)\ldots\left(x - w^{n - 1}\right) $$ Prove that: $$ \sin\left(\pi \over n\right)\sin\left(2\pi \over ...
0
votes
2answers
43 views

$f$ is an entire function satisfying the given condition . Show that the function is constant

If an entire function $f(z)$ satisfies $$|f(z)|\le \frac{1+|z|}{1+\sqrt {|z|}}$$ for all $z\in \mathbb C$ then show that $f=c$ with $|c|\le 2(\sqrt 2-1)$. First we consider a function ...
1
vote
0answers
56 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
1
vote
0answers
38 views

Proving that two or three segments are concurrent using complex numbers or vectors.

For example if we have a triangle and we want to prove that the medians all intersect at a point, using complex numbers (or vectors); how do we do that? (This is not my main question) My problem is ...
1
vote
1answer
31 views

Find all the complex numbers that satisfy this quotient.

A certain problem that I have been working on involves the equation $$1 = \frac{1}{1-n}$$ One can see that the only real-number solution is $n=0$. As far as the original problem goes, that is ...
0
votes
5answers
123 views

Why is $\sqrt{-x}*\sqrt{-x}=-x?$

Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q2 - I was thinking it would be: $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?) Q3 - What are the formal algebra rules to use? Can ...
-3
votes
1answer
55 views

Calculating a complex number

From some reasons (trying to solve the cubic equation $11925\,{z}^{3}-1219\,{z}^{2}-19186\,z+360=0$ with positive discriminant) I know that the number $$ a=\sqrt [3]{201401326+12825\,i\sqrt ...
3
votes
4answers
291 views

Obtain real part of complex expression

I must verify if the real part of the following expression $$z = \frac{1 + i}{\sigma \delta \left[ 1 - e^{-(1 + i)t/\delta} \right] }$$ is $$\Re(z) = \frac{1}{\sigma \delta} \frac{1}{1 - ...
2
votes
2answers
67 views

How many analytic functions are there satisfying the given condition?

How many analytic functions $f(z)$ are there in $\Omega$ with the property that $f(z)^2+3if(z)+4\equiv 0$ on $\Omega$ ? where, $\Omega$ is the whole complex plane with two co-ordinate axes ...
3
votes
1answer
28 views

Evaluating a complex integral using resiue theorem

Evaluate the integral $$\int_{|z+1|=2} \frac{z^2}{4-z^2}dz$$ Solution : So $|z+1|=2$ is the circle of radius 2 centered at -1. Now inside this circle $\frac{z^2}{4-z^2}$ is analytic except for a ...
1
vote
5answers
239 views

Cardinality of a set of complex numbers

The question is basically to find the number of elements in the set $\{z \in \mathbb{C} : z^{60} = -1 , z^k \neq -1, 0<k<60 \}$. As is quite obvious with the kind of question,I am a ...
0
votes
2answers
63 views

Polynomial with exactly one complex root

Is it possible that a polynomial of degree $n$ with real coefficients has exactly one complex root? I saw https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem but wondered if this can happen ...
4
votes
1answer
55 views

Inverse image of $[-2,2]$ under cosine.

I solved the following problem: Let $g(z) = \cos z$. Find $g^{-1}[-2,2]$. but my solution was kind of long. I was wondering if there was a faster way to do this problem. Here's my solution: ...
2
votes
2answers
36 views

Find all the solutions of the equation $w^3 = 1/2(1+i\sqrt{3})$ in the form $r.\operatorname{cis}(\theta)$

Could anyone please help me walk through the steps. I understand the underlying concepts through the use of deMoivre's Formula, and that it is to be written in the form of Euler's equation. I would ...
1
vote
5answers
139 views

Why is $-i^3 = i$?

Why is the value of $-i^3$ equal to $i$? After experimenting, I got this result - $-i^3=-i^2\cdot -i=1 \cdot -i=-i$ What is the error in my proof? EDIT Here is the original proof - ...
9
votes
2answers
144 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...