What is Complex algebra: Definition and 22 Discussions
In mathematics, a field of sets is a mathematical structure consisting of a pair
⟨
X
,
F
⟩
{\displaystyle \langle X,{\mathcal {F}}\rangle }
consisting of a set
X
{\displaystyle X}
and a family
F
{\displaystyle {\mathcal {F}}}
of subsets of
X
{\displaystyle X}
called an algebra over
X
{\displaystyle X}
that contains the empty set as an element, and is closed under the operations of taking complements in
X
,
{\displaystyle X,}
finite unions, and finite intersections.
Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over
X
{\displaystyle X}
" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
The Youtuber 'Blackpenredpen' claims that ##1^x=2## has solutions ##x= \frac{-i \ln(2)}{2\pi n}## with ##n \in \mathbb{Z}## and ##n \neq 0##. Somone else in a forum claims that because ##1^x## is not injective there are more solution branches and this solution mixes these branches somehow. Who...
Problem Statement: What is the correct way of computing the argument of the following equation?
Relevant Equations: I am trying to compute the argument ##\Phi## of the equation
$$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$
which using Euler's equation...
Homework Statement
Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
Homework Equations
$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$
The Attempt at a Solution
First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
Homework Statement
Write ##5-3i## in the polar form ##re^\left(i\theta\right)##.
Homework Equations
$$
|z|=\sqrt {a^2+b^2}
$$
The Attempt at a Solution
First I've found the absolute value of ##z##:
$$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$.
Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
Homework Statement
Show that
$$\int_C e^zdz = 0$$
Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i.
Homework Equations
$$z = x + iy$$
The Attempt at a Solution
I know that if a function is analytic/holomorphic on a domain and the contour lies...
Homework Statement
A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E.
a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1.
b) Calculate the probability current density...
The problem
I would like to solve:
$$ \bar{z} = z^n $$ where ##n## is a positive integer.
The attempt
## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ##
## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1##
##...
I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { }
(This question is probably asked a lot but I could not find it so I'll just ask it myself.)
Does the square root of negative numbers exist in the complex field? In other words is...
Homework Statement
I just can't seem to get the right answer. z^4+80i=0
looking at the complex plane u see the radius=r=80 (obviously)
using De Moivre extension: z^n=(r^(1/n))(cos((x/n)+k2pi/n)-isin((x/n)+k2pi/n)z1=((80)^(1/4))(cos(3pi/8)+isin(3pi/8)
shouldnt this be a root?z2=...
ok having major problems. i can easily solve z^2 + pz +a+bi=0 solutions but that extra qiz is really annoying me.
z^2 + 3z+4iz-1+5i=0
(z+2i)^2+3z-5+5i=0
z+2i = w, z=w-2i
w=-3(w-2i)+5-5i
then I am not getting anything sensible for solving x and yi. what am i doing wrong?
Homework Statement
First off i wasn't sure if i should put this in precalc or here so i just tossed a coin[/B]
I must find the roots of the expression z^4 +4=0 (which I've seen repeatedly on the internet)
Use it to factorize z^4 +4 into quadratic factors with real coefficients
The answer is...
I'm trying to understand something in my notes here...
So if we call the real part of the complex algebra 'even' and the imaginary part 'odd' then this graded algebra is communitive but NOT graded commutative. so ab = ba for all a and b in C.
If we call the whole complex algebra 'even' and...
Homework Statement
This is one part of a wider question, I'm only posting the part I'm having trouble with.
$$
\begin{align}
\text{Given an impedance network } B &= \frac{Z_1 \parallel Z_3}{Z_2 + Z_1 \parallel Z_3} \\
\text{show that: } \frac{1}{B} &= 1 + \frac{R_2}{R_1} + j\frac{\omega CR_2}{1...
I've attached the question that I am referring to.
I believe I'm heading in the right direction with this one by stating that:
1/Rt = 1/(6+j8)Ω + 1/(9-j12)Ω
But I am confusing myself with my algebra.
Any help is appreciated
Homework Statement
https://wiki.math.ntnu.no/lib/exe/fetch.php?hash=d26b1f&media=http%3A%2F%2Fwww.math.ntnu.no%2Femner%2FTMA4115%2F2012v%2Fexams%2Fkont.eng.pdf
Assignment 1.
"Find all complex numbers s such that Im(-z + i)= (z+i)2"
What do I do?
Homework Equations
The Attempt at a...
Homework Statement
Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values
Find all the values in the complex plane for which |tan(q)| = ∞
Homework Equations
Expressing tan(q) as complex exponentials:
(e^iq - e^(-iq))/i(e^iq + e^(-iq))
The Attempt...
Homework Statement
\frac{\beta}{\alpha + i 2\pi f} = \frac{\alpha \beta}{\alpha^{2}+ 2 \pi f}- i \frac{2 \pi \beta}{\alpha^{2}+ 2 \pi f}
Homework Equations
complex conjugate:
(a + ib) * (a - ib) = a^2 + b^2
The Attempt at a Solution
If it matters, this is from a book on Fourier...
Homework Statement
I am told: {\frac {{\it du}}{{\it dx}}}=y and {\frac {{\it du}}{{\it dy}}}=x. Need to find u(x,y) which is a real valued function and prove the result.
Homework Equations
The Attempt at a Solution
Well, I think the answer is of the form u(x,y) = xy + c because...
The question reads:
"what part of the z-plane corresponds to the interior of the unit circle in the w-plane if
a) w = (z-1)/(z+1) b) w = (z-i)/(z+i)"
I really am having problems understanding what the question is asking. I don't understand what the w plane is, and in which plane...