Complex algebra Definition and 22 Threads

I was reading a book on complex numbers where I stumbled upon this article. I was following it when I found some error. The distance between the center of circle and line should be mod(z star - z1) right? Z0 is just any random pt on line. They have continued this fallacy in example 3.10...

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...

Are there any mathematical models which operate with physical time as with hypercomplex number? If yes, are there any related experiments?

Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$

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 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...