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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7
votes
8answers
984 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 ...
0
votes
1answer
91 views

Multiplicities of zeros of the polynomial $p(z)=1+2z^4+\frac 7 {10}z^{10}$

How many zeros does $p(z)=1+2z^4+\frac{7}{10}z^{10}$ have in the unit disc ? Determine multiplicities of these zeros. Let , $f(z)=2z^4$ and $g(z)=1+\frac{7}{10}z^{10}$. Then, on $|z|=1$ , ...
2
votes
3answers
74 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 ...
0
votes
1answer
60 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
4
votes
1answer
117 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 ...
2
votes
2answers
214 views

Show that $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ represents a circle

Suppose $|a|<1$ and $r\in (0,1)$. Show that the set of complex number $z$ satisfying $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ is a circle in complex plane. Find the centre and radius of this ...
0
votes
2answers
82 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
82 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
1answer
56 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 ...
12
votes
2answers
199 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
19 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}]} ...
0
votes
5answers
131 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
59 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 ...
2
votes
2answers
78 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
4answers
308 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 - ...
3
votes
1answer
33 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
294 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 ...
1
vote
0answers
76 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 ...
0
votes
2answers
85 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
59 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: ...
1
vote
5answers
144 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 - ...
4
votes
2answers
152 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 ...
1
vote
2answers
41 views

Polar exponential form of -i

Express $-i$ in form $re^{i\theta}$ $r=1$ is simple enough. As on an Argand diagram $-i$ will be at $(0,-1)$ is $\theta = 3\pi/2$ here? or $-\pi/2$ to get it $-\pi < \theta < \pi$ Is the ...
-3
votes
2answers
67 views

Equilateral triangle in complex number [closed]

Let $a$, $b$ and $c$ be the affix of $A$, $B$ and $C$, where $a+b+c=0$ and $a$, $b$ and $c$ are of equal magnitude. Prove that $\triangle ABC$ is an equilateral triangle.
-4
votes
1answer
102 views

Express -i in polar exponential form

Express $-i$ in form $r\cdot e^{i\cdot \theta}$ $r=1$ is simple enough. As on an Argand diagram, $-i$ will be at $(0,-1)$. Does $\theta = 3\pi/2$ here? Or -$\pi/2$ to get it $-\pi < \theta < ...
2
votes
2answers
38 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 ...
2
votes
3answers
39 views

$\varphi_n (z) = z^n$ find all subgroups between $\ker \varphi _3$ and $ \ker \varphi _{12}$

let $C^*$ be the group of complex numbers excluding zero with * operation. I need to show the following - 1) $\varphi_n : C^* \to C^*$ such that $\varphi_n (z) = z^n$ is homomorphism. 2) $\ker ...
0
votes
2answers
110 views

Finding nth roots of complex number [duplicate]

If I have a complex number of the form $z = a+bi$, how would I find the complex roots? I know that each root will be equidistant from each other and will form a circle, but I'm not sure how to solve ...
0
votes
1answer
55 views

Complex numbers simplify

I'm new here and I'm studying complex numbers is there a way the simplify this: $$\left(\frac{45\sqrt{3}i}{i^\pi}\right)^{\pi \times 5}$$
2
votes
5answers
264 views

Finding argument of complex number and conversion into polar form

How do I find the argument of a complex number, for example $z = 3 + 4i$? I know the polar form of $z$ is $r(\cos\theta + i\sin\theta)$ where $r$ is the modulus of $z$ ($\sqrt{3^2+4^2}$) which would ...
0
votes
3answers
82 views

Complex numbers confusion

I am a little bit confused trying to understand complex numbers. I read Richard Feynman lectures on physics and in chapter about complex numbers he says: 10^is = x + iy | i - imaginary, s - real ...
1
vote
1answer
59 views

how to compute $|a-ib|^2$ if given $(a-ib)^3$

If i know that, for example, $(a-ib)^3=5+4i$ - how can i compute the value of $|a-ib|^2$ ? I can take modulus of $5+4i$ which is $\sqrt{5^2+4^2}$ but i don't know what i'm getting here. I don't ...
1
vote
3answers
182 views

Composition of Rotation and Translation in the Complex Plane — Finding Angle of Rotation and Point

A rotation about the point $1-4i$ is $-30$ degrees followed by a translation by the vector $5+i$. The result is a rotation about a point by some angle. Find them. Using the formula for a rotation in ...
2
votes
2answers
146 views

When does $i^x=x$

Can someone please help me solve $i^x=x$? So far I have: $$i^x=x$$ $$\frac{\ln(x)}{\ln(i)}=x$$ $$e^{i\pi}=-1$$ $$e^{i\pi/2}=i$$ $$\frac{\ln(x)}{\frac{i\pi}{2}}=x$$ $$\ln(x)=\frac{i x \pi}{2}$$ ...
3
votes
1answer
70 views

Simplify expression involving real or imaginary part of complex rational function

Basically I want simplify the following so that the real or imaginary operator do not appear: $$\Im \prod_{i=1}^{N-1} \left( z-x_i\right)^{l_i}$$ or $$\Re \prod_{i=1}^{N-1} \left( ...
1
vote
4answers
66 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 ...
0
votes
1answer
27 views

set of points of $M$ the affix of $Z$

let M be the affix of $Z$ Determine the set of positions of $M$ the affix of $Z$ when the images of the numbers $1,Z$ and $Z^2+1$ are collinear. I did $\arg((1-z)/(1-1-z²))=0$ then ...
1
vote
3answers
89 views

If x and y are real, solve the equation $\frac{xi}{1+yi}=\frac{3x+4i}{x+3y}$

If x and y are real, solve the equation $$\frac{xi}{1+yi}=\frac{3x+4i}{x+3y}$$ I have tried giving both sides of the equation a common denominator of $(1+yi)(x+3y)$ and then manipulating the ...
2
votes
2answers
43 views

Quadratic Polynomial with complex coefficients

Let polynomial $p(z)=z^2+az+b$ be such that $a$and $b$ are complex numbers and $|p(z)|=1$ whenever $|z|=1$. Prove that $a=0$ and $b=0$. I could not make much progress. I let $z=e^{i\theta}$ and ...
1
vote
4answers
120 views

Real part of $\frac{1-e^{(n+1)i\theta}}{1-e^{i\theta}}$

How can I compute the real part of \begin{equation*} \frac{1-e^{(n+1)i\theta}}{1-e^{i\theta}}, \quad \text{where}\ \theta \in \mathbb{R}? \end{equation*} Maybe it's a silly question, but I'm feeling ...
1
vote
1answer
61 views

Where's the flaw in this application of De Moivre's fomula to find n-th roots?

To find the $n^\text{th}$ roots of a complex number, we can first express it in polar form (I'm assuming $r=1$ for brevity; it doesn't matter for my question): \begin{align} e^{i\theta} &= ...
1
vote
1answer
56 views

Determine $Z(f)=\{z:f(z)=0\}$

Let, $f(z)=\cos(iz^3)$. Determine $Z(f)=\{z:f(z)=0\}.$ Indicate with a picture where the solutions lie in $\mathbb C$. $f(z)=0$ gives, $\cosh(z^3)=0\implies e^{2z^3}=-1=e^{(2k+1)\pi}\implies ...
3
votes
1answer
122 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 ...
0
votes
1answer
15 views

Evaluate and simplify multiplication of exponents with base e; polar forms

$$2e^{(i×\pi/4)}×3e^{(i×\pi/6)}$$ How would I evaluate and simplify the above, and then express it in polar form? I understand $re^{i\theta} = r(\cos\theta+i\,\sin\theta)$. The question is to find ...
3
votes
0answers
67 views

Solving simultaneous equations in complex numbers

Given $z_1,z_2$ are complex numbers such that sum of their squares is a real number and $$z_1(z_1^2-3z_2^2)=2$$ and $$z_2(3z_1^2-z_2^2)=11.$$ I need to find the value of sum of squares of two complex ...
3
votes
2answers
235 views

What is the bar symbol over a complex scalar in the expression $\overline{\lambda}$?

I have the following problem from section 1.4 (Vector Spaces) of Peter Peterson's Linear Algebra textbook. I am having trouble with the way multiplication is defined on the given vector space, ...
0
votes
5answers
58 views

Roots of Unity: second largest value and absolute value

Consider the $n$th roots of unity $e^{2 \pi i k/n}$ for fixed integer $n \geq 2$ and $0 \leq k < n$. Now I am interested in the second largest value (in absolute value) of the values ...
1
vote
1answer
91 views

Rotation around complex number

The function $$f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$$ represents a rotation around some complex number $c$. Find $c$. How would I start this? Thanks.
0
votes
2answers
55 views

Find integer solution of sysem of quadratic equations [closed]

If: $a,b,c$ positive integers, where $a\geq b\geq c$. such that: $$a^2 - b^2 - c^2 +ab=2011$$ $$a^2 +3b^2 +3c^2 -3ab-2ac-2bc=-1997.$$ Find the value of $a$ I tried, but I got nothing. Source: 2012 ...
1
vote
3answers
105 views

Application of chain rule for a complex variable

I'm looking at basic definitions in complex analysis, and I can't figure out where a factor of $1/2$ comes from below. All sources I've found either invoke it without explanation, or derive it after ...