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

learn more… | top users | synonyms

3
votes
1answer
25 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
234 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
36 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
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: ...
3
votes
5answers
132 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
141 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 ...
2
votes
2answers
39 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
46 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
54 views

Express -i in polar exponential form

so 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 ...
4
votes
2answers
34 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
35 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
52 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 ...
2
votes
1answer
32 views

Complex numbers simplify

I'm new here and I'm studying complex numbers is there a way the simplify this: $$\left(\frac{45\sqrt{3}i}{i^\pi}\right)^{\pi \times 5}$$
3
votes
5answers
88 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
66 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
46 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 ...
0
votes
3answers
17 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 the ...
2
votes
2answers
131 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
19 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
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 ...
0
votes
1answer
21 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
46 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
31 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 ...
2
votes
5answers
79 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
46 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
53 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
98 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
7 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
37 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 ...
0
votes
1answer
35 views

A function built geometrically

A function $f$ of the disk $| z | < 1$ in $\mathbb {C}$, is defined as follows: Let $z=OP$ be in the disk. 1) it is drawn the perpendicular to the segment $OP$ at the point $P$ which cuts the ...
3
votes
2answers
36 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
29 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
53 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
54 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
64 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 ...
0
votes
1answer
38 views

Find vertices of equilateral triangle

There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $2 + 2i$, $5 + i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1 c_2$. So far, I have used the distance ...
0
votes
1answer
43 views

Intersection of two lines in complex numbers given four points [closed]

How to find the point of intersection of two lines, given four points, two of which are on each line, in complex numbers? Thank you!
1
vote
1answer
50 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
2
votes
2answers
76 views

Role of i in Fourier transform

I've seen several derivations of the Fourier transform, but most don't cover the conversion to the form $$ S(f) = \int_{\infty}^{-\infty} s(t)e^{-i2\pi ft} \;\mathrm{d}t $$ What is the role of ...
0
votes
0answers
11 views

Error on determinant from statistical errors on complex matrix elements

Say I have a complex matrix $A$ whose elements $A_{ij}$ have statistical error $\delta_{ij}$. I need to figure out from these errors what will be the error on the determinant $|A|$. If the matrix was ...
1
vote
2answers
48 views

Plotting the equation $x - {(\cos(x) + i\sin(x))^{ix}} = 0$

I would like to plot this function: $$x - {(\cos(x) + i\sin(x))^{ix}} = 0$$ I remember about $\cos(x) + i\sin(x) = e^{ix}$, so this can be written as $$x - {(e^{ix})^{ix}} = 0$$ or maybe better $$x ...
2
votes
1answer
38 views

Application of Luca's theorem

Let, $p$ be a polynomial in $1$-complex variable. Suppose all zeros of $p$ are in the upper half plane $H=\{z\in \mathbb C|\Im(z)>0\}. $ Then , which are corrct ? ...
5
votes
4answers
78 views

Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
1
vote
3answers
217 views

Find the value of $x$ such that $\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$

Find the value of $x$, $$\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$$ Help guys please, I have tried and I got, $x=-2, x=1$, and I think it's wrong
-2
votes
1answer
36 views

Consider the square defined by $0 \leq Re(z) \leq 1$ and $0 \leq Im(z) \leq1$. Determine … [closed]

I have the following problem to solve: Consider the square defined by $0 \leq \operatorname{Re}(z) \leq 1$ and $0 \leq \operatorname{Im}(z) \leq1$. Determine the image of this square by the ...
2
votes
3answers
58 views

Is there a simple way to define the $n$-th roots of the unity?

Is there a simple way to calculate the $n$-th roots of the unity? I gotta solve the equation $$\frac{z+1}{z-1}=\sqrt[n]{1}.$$
1
vote
3answers
231 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
0
votes
2answers
33 views

Condition for the argument when complex numbers are written in polar form

In my text book it says that the complex number z(not equal to 0) can be written in polar form as $z = r(\cos\theta + i \sin\theta)$, where r = mod z greater than 0 is the modulus and $\theta = \arg ...
1
vote
1answer
55 views

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...