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

0
votes
1answer
34 views

An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$ |z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2 $$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!
1
vote
2answers
64 views

Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
0
votes
0answers
18 views

Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
0
votes
3answers
65 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
0
votes
0answers
39 views

Complex Matrix Representation

Lets say if $X\in C ^{m\times n}$, it does have real and imaginary parts. If I want to represent a matrix in real and imaginary form then why I write it this way where is $i$? \begin{bmatrix} X_r ...
1
vote
0answers
36 views

Proving analytic function $f = 0$ under certain assumtions

I was given the following exercise: Let $f(z)$ be analytic in an open and connected set $U$ containing the point $z=0$ and assume $|f(1/n)| < \frac{1}{2^n}$ for $n \in \mathbb{N}_{> 0}$. Prove ...
0
votes
2answers
29 views

How to find the absolute value of this complex number: $\frac{-4-6i}{17+i}$

I know that, in general, $|a+bi|=\sqrt{a^2+b^2}$, however, I don't know how to make $\frac{-4-6i}{17+i}$ into the form of $a+bi$.
0
votes
1answer
39 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
0
votes
0answers
18 views

Solving systems of linear equations with complex numbers by hand

How can I solve a 3x3 system of linear equations with complex numbers by hand without making a mistake? I know that I can solve them either with Gaussian Elimination or Cramer's rule, but I find it ...
2
votes
1answer
43 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle ...
3
votes
1answer
67 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha ...
2
votes
1answer
46 views

Simplifications of nth roots of complex numers.

Is it easy to find that $( a + ib)^n$ is equal to a certain complex number, say $p+iq$, by just using Newton's binomial theorem. But how to find in general that $\sqrt[n]{p+iq} = a+ib$, where $a, b, ...
1
vote
1answer
56 views

Does $z^0$ Have Multiple Solutions?

While playing around with complex numbers, I stumbled upon a result that implies $z^0$ has infinitely many values (where $z$ is any complex number). This struck me as odd since I've never come ...
-1
votes
1answer
26 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
1
vote
4answers
36 views

Complex numbers, finding solution for z

How can you solve this? $z^2+2(1+i)z=2+2(\sqrt{3}-1)i$ I have tried to compare left and right side with real and imaginary part i then get $ x^2+2x-y^2-2y=2$ $xy+x+y=(\sqrt{3}-1)$ But this ...
1
vote
1answer
56 views

Why don't we have $(\mathrm{cis}(2\pi))^{1/5} = (\mathrm{cis}(4\pi))^{1/5}$, while we do have $\mathrm{cis}(2\pi) = \mathrm{cis}(4\pi)$?

If $\operatorname{cis}(2\pi) = \operatorname{cis}(4\pi)$, then don't we have $$\big(\operatorname{cis}(2\pi)\big)^{1/5} = \big(\operatorname{cis}(4\pi)\big)^{1/5}?$$ This isn't yielding the same ...
-2
votes
3answers
54 views

Converting into rectangular form

I have 2 related questions: First: Let $z_1 = 2+2i$ and $z_2 = 2-2i$. Find $z_1z_2 $ in rectangular form. I have no idea... I'm also clueless about this question: Change the following to ...
0
votes
0answers
18 views

Graphing complex numbers on Casio fx-9750 GII

I have a Casio fx-9750GII, I'm starting to get to grips with just how useful it can be. For those familiar with it you will be aware it can graph functions. However I noticed when looking through the ...
0
votes
1answer
23 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
3
votes
1answer
35 views

Show that $\xi^3\equiv \pm 1 \pmod{\lambda^4}$ in $\Bbb Z [\omega]$

We have $\lambda=1-\omega$ where $\omega=e^{i 2\pi/3}$ and $\xi$ an Eisenstein integer. Given that $\xi \equiv \pm 1 \pmod{\lambda}$, how can I prove that $$\xi^3\equiv \pm 1 \pmod{\lambda^4}$$ I ...
1
vote
1answer
34 views

Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
0
votes
0answers
43 views

Find a Möbius transformation $g$ such that $f(z) = g(z)$ for all $z \in D$.

Let $f$ be a function with domain $D= ${$z \in \mathbb C$: $|z| < 8$} given by $$ f(z) =\sum_{n≥0}\dfrac{i^nz^n}{8^n}. $$ Find a Möbius transformation $g$ such that $f(z)=g(z)$ for all $z \in D$. ...
2
votes
2answers
39 views

How to go from $\frac{1}{1+2j}$ to $\frac{1}{5} - \frac{2}{5}j$, where $j^2=-1$?

I am reading a book (DSP First), or mainly skipping through the pages trying to solve various exercises. At some point I came across this How exactly did we go from the second to the last step?
2
votes
0answers
36 views

Tetration of a number giving a complex number

Giving this power equation: $$S=\lim_{n\to\infty} {^n}x=-i$$ where the symbol $^nx$ means the tetration operator, we can write in a form not formally correct: $${\ ^{n}x = \ \atop {\ }} ...
0
votes
4answers
38 views

Equation with powers of complex variables and its conjugate

How do you solve an equation like this? $(3z)^2 + (4\bar{z})^4 = 0 $ I tried setting the complex variable to its polar form $re^{i\theta}$ and simplifying but ended up with $re^{-i3\theta} = ...
0
votes
0answers
11 views

Is this factorization of $\xi^3 \mp 1$ in $\mathbb{Z}[\omega]$ correct?

I'm trying to follow a proof of Fermat's Last Theorem for $n=3$ using the Eisenstein integers according to this paper. On page $6$ near the bottom the author gives the factorization $$\xi^3 ...
3
votes
2answers
74 views

There is no 'nice' complex logarithm

Two problems are meant to establish that no 'nice' logarithm function exists for complex numbers. The first is Let $U$ be an open set in $\mathfrak{C}\setminus\{0\}$. Suppose $h:U \to \mathfrak{C}$ ...
3
votes
2answers
43 views

Given $f(z)=\dfrac{U(z)}{V(z)}=\dfrac{2z^3-3z^2+7z-8}{z^4-5z^3+4z^2-6z+1}$ find $f(1-\sqrt{2}i)$ without lots of complex arithmetic.

Such a problem is usually done either by direct substitution (ugh!) or synthetic division. Synthetic division after several complex products and additions gives $U(1-\sqrt{2}i)=-8-3\sqrt{2}i$ After ...
1
vote
1answer
17 views

How to “simplify” parallel vectors in $\mathbb{C^3}$?

Sorry for the bad title but I can't find a better one. $\vec{v}=(v_1,v_2)^T\in\mathbb{R^2}$ $-\vec{v}=(-v_1,-v_2)^T$ For my needs $\vec{v}$ and $-\vec{v}$ are equivalent. In my problem $\vec{v}$ is ...
0
votes
0answers
26 views

Complex conjugate of the logarithm of the hyperbolic tangent

Given the Schwarz reflection principle, I would aytomaticaly write down that the complex conjugate of the following function: $$ ln[tanh(z)] $$, where z is a complex number, is: $$ln[tanh(\bar{z})] ...
0
votes
2answers
20 views

Square roots of complex number in exponential form

The complex number $z$ is defined by $z=\frac{9\sqrt{3}+9i}{\sqrt{3}-i}$. Find the square roots of $z$, giving your answers in the form $re^{i\theta}$.where $r>0$ and $-\pi < \theta \leq \pi$. ...
0
votes
1answer
30 views

Give a conformal map, with certain initial conditions, from the open unit disc to another open set …

... that open set being $$\mathbb{C} - \{x\in\mathbb{R} : x\leq-\frac{1}{4}\}$$ and the boundary conditions being $f(0) = 0$ and $f'(0) = 1$. Here is my first try and only idea so far: $Ci\frac{1 - ...
6
votes
1answer
173 views

Why can't we order Complex Numbers? [duplicate]

I know this may very well be a silly question. I always hear that Complex numbers cannot be ordered. But there's something I'm missing... Why can't we just compare two complex numbers $z_1,z_2$ as ...
0
votes
0answers
25 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} ...
2
votes
0answers
40 views

Does the exponent property $(a^{m})^{n} = a^{mn}$ break down in Real Analysis

Preface: I recently asked a question (The Definition of the Absolute Value), on the definition of the absolute value, in which I used $|x| = \sqrt{x^2}$, along with the exponent property $(a^{m})^{n} ...
1
vote
1answer
56 views

Complex numbers and circles

Question: Let the complex numbers $\alpha$ and $1\over \bar\alpha$ lie on circles $$(x-x_0)^2 + (y-y_0)^2 = r^2$$ and $$(x-x_0)^2 + (y-y_0)^2 = 4r^2$$ respectively. If $z_0 = x_0 + iy_0$ ...
0
votes
2answers
19 views

Maple complex exponential form to algebraic.

How can I easily convert complex number from exponential or trigonometric form to algebraic? Update In fact I'm trying to simplify this expression: The only way I see is to convert to ...
1
vote
1answer
39 views

How can this Complex problem?

The problem is $e^z=i+\sqrt{3}$. I put $z= x+yi$ ,then $i+\sqrt{3}$ is $2(\cos\left(\frac \pi 6\right)+\sin\left(\frac \pi 6\right)\cdot i)$. So $e^x$ is $2$ and $y$ is $\frac \pi 6+2k\pi$. ($k$ is ...
-1
votes
1answer
30 views

What is the domain and codomain of a transfer function? [closed]

Let's say I have the transfer function- $\textbf{H}(j\omega)=\cfrac{1}{1+j\omega RC}$ Where does this function map to and from, and can it be plotted visually?
1
vote
1answer
34 views

Is the space $\mathbb{R}_+\times S\times S$ linear?

The space $\mathbb{C}$ (or even $\mathbb{R}^2$), which is a linear space over $\mathbb{R}$, can be obtained from the Cartesian product $\mathbb{R}_+\times S$ by gluing to the point the layer $0 ...
0
votes
1answer
12 views

For complex number $z_1$ and $z_2$ determine angle $z_1 O z_2$

Let $w=\frac{\sqrt{3}+i}{2}$ and $P=\{w^n:n=1,2,3,.... \}$ Further $H_1=\{z \in C: Re(z)>\frac{1}{2} \}$ and $H_2=\{z \in C: Re(z)<-\frac{1}{2} \}$, where $C$ is the set of all complex numbers. ...
0
votes
1answer
18 views

Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate?

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} ...
1
vote
1answer
16 views

Indicate on an Argand Diagram the region of the complex plane in which $ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $

Question: Indicate on an Argand Diagram the region of the complex plane in which $$ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $$ I've tried this Consider $$ 0 \leq \arg (z+1) \leq ...
1
vote
0answers
38 views

Number of solutions of a complex equation

Question: Find the number of solutions of the equation: $$z^3 + \frac{3(\bar z)^2}{|z|} = 0$$ I substituted $z = re^{i\theta}$ to convert the equation into: $$r^3e^{i3\theta} + 3re^{i2\theta}=0$$ ...
1
vote
1answer
21 views

Solving for complex power

I am asked to find $(1+i)^{3+4i}$ This is what I have and I wanted to know if it is correct: $e^{3+4i [ln\sqrt{2} + i (\pi/ 4 +2\pi k]}$ $e^{3+4i ln\sqrt{2} + 3-4 (\pi/ 4 +2\pi k)}$ by ...
3
votes
2answers
48 views

Closed contour integral: $\int_{\mathbb{c}}\frac{ z}{2z^{2}+1} dz$ where the contour is the unit circle

first and foremost please excuse my English. given $∫_c \frac{{z}}{2z^{2}+1}dz$ where the contour is the unit circle. so c = $e^{it}$ from 0 to $2\pi$. since the contour is the unit circle we can ...
0
votes
2answers
59 views

What is the trigonometric form of the complex variable $z=0+0i $?

I'm confused how do i determine the trigonometric form of the complex variable $z=0+0i$ , it has modula such that is 0 but what about it's argument ? Note : At a least i would like to know it's ...
77
votes
7answers
4k views

Why are There No “Triernions”? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
0
votes
2answers
26 views

Complex number and polar coordinates

True or false: the polar coordinates of $-1-i$ are $-\sqrt{2}\operatorname{cis}\frac{\pi}{4}$ In my opinion it's true: $\tan\theta=\frac{-1}{-1}=1\Rightarrow \theta=\frac{5\pi}{4}$, ...
2
votes
2answers
40 views

Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...