1. ### I Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices

I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...
2. ### Exponential Wavefunction for Infinite Potential Well Problem

Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...
3. ### Find the set of points that satisfy:|z|^2 + |z - 2*i|^2 =< 10

Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...
4. ### I The domain of the Fourier transform

Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...
5. ### I Principal difference between complex numbers and 2D vectors revisited

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...

19. ### How do you work out simultaneous eqns w/ complex numbers & phasor

I'm having trouble figuring out to get the answers from the 2 equations. The phasors and complex numbers confuse me. Do I need to change the phasor form? How do I go about doing this thanks! (Not homework question im trying to figure this for my exam!)
20. ### Stuck on expressing a complex number in the form (a+bi)

Homework Statement Express the complex number (−3 +4i)3 in the form a + bi Homework Equations z = r(cos(θ) + isin(θ)) The Attempt at a Solution z = -3 + 4i z3 = r3(cos(3θ) + isin(3θ)) r = sqrt ((-3)2 + 42) = 5 θ = arcsin(4/5) = 0.9273 ∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273)) a = -117 b...
21. ### Nonzero complex numbers

Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework Equations The Attempt at a Solution I...
22. ### Complex numbers. write equation on form "a+bi"

Homework Statement Write this complex number in the form "a+bi" a) cos(-pi/3) + i*sin(-pi/3) b) 2√2(cos(-5pi/6)+i*sin(-5pi/6)) Homework Equations my only problem is that im getting + instead of - on the cosinus side.(real number) The Attempt at a Solution a) pi/3 in the unit circle is 1/2 for...
23. ### Complex numbers - factors

The problem The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions? The attempt This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...
24. ### A Basic Spectral Analysis proof help

I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
25. ### Complex polynomial help

Homework Statement [/B] Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero. The Attempt at a Solution [/B] (z-(1+i)(z-i) = Z^2-z-1-2iz+i (Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i) Z^2 term...
26. ### Complex numbers in the form a+bi

Homework Statement How would I go about solving 1/z=(-4+4i) The answer that I keep on getting is wrong The Attempt at a Solution [/B] What I did z=1/(-4+4i)x(-4-4i)/(-4-4i) z=(-4-4i)/(16+16i-16i-16i^2) z=(-4-4i)/32 z=-1/8-i/8 This is the answer that I got however it says that it is...
27. ### How to find the third root of z^3=1?

Homework Statement in a given activity: solve for z in C the equation: z^3=1 Homework Equations prove that the roots are 1, i, and i^2 The Attempt at a Solution using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2) it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution...
28. ### Complex numbers on the unit circle

Homework Statement Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle. Homework Equations The Attempt at a Solution As far as I understand, I need...
29. ### I Proving De Moivre's Theorem for Negative Numbers?

Or basically anything that isn't a positive integer. So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds. If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)## My proof below: However I struggle to do this with...
30. ### B Complex Integration By Partial Fractions

Hello, I am enrolled in calculus 2. Just having started a section in our text book about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...