# What is Complex: Definition + 1000 Threads

The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) together form the UCL School of the Built Environment, Engineering and Mathematical and Physical Sciences.

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1. ### Are these values of the following correct? (Complex cube root of unity)

Given that x is a complex cube root of unity, we have ## x^{3}=1 ## but ## x\neq 1 ##, where the cube roots of unity are ## 1, \omega, \omega^2 ## and ## \omega, \omega^2 ## are the imaginary roots such that ## \omega^3=1, 1+\omega+\omega^2=0 ##. Now we will consider three cases of the...
2. ### I Question about partial derivative relations for complex numbers

Apologies this is probably a very bad question but it's been a while since I have seen this. I have ##z=x+iy##. I need to convert ##\frac{\partial \psi(z)}{\partial z}## , with ##\psi## some function of ##z##, in terms of ##x## and ##y## I have ##dz=dx+idy##. so ##\frac{\partial \psi }{\partial...
3. ### POTW Cohomology Ring of Complement of Hyperplanes in Complex n-space

Let ##n \ge 1## and ##1 \le m \le n##. Suppose ##\lambda_1,\ldots, \lambda_m : \mathbb{C}^n \to \mathbb{C}## are ##\mathbb{C}##-linearly independent linear functionals. For each abelian group ##G##, determine the cohomology ring $$H^*(\mathbb{C}^n \setminus \bigcup_{k = 1}^m \ker \lambda_k ; G)$$
4. ### Textbook 'The Physics of Waves': Derive Complex Solution Form for SHM

TL;DR Summary: Question on deriving the complex irreducible solution form for simple harmonic motions based on time translation invariant. Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 1 -...
5. ### Insights Views On Complex Numbers

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### Fundamental matrix for complex linear DE system

For this problem, I am trying to find the fundamental matrix, however, the eigenvalues are both imaginary and so are the eigenvectors. That is, ##\lambda_1 = 4i, \lambda_2 = -4i## ##v_1 = (1 + 2i, 2)^T## ##v_2 = (1 - 2i, 2)^T## So I think I just have an imaginary matrix? This is because the...
7. M

### Complex Eigenvalues of system of DE

For this problem, Can someone please explain to me how they got from the orange step to the yellow step? I am confused how the two expressions are equivalent. Thanks!
8. ### A What causes a complex symmetric matrix to change from invertible to non-invertible?

I'm trying to get an intuitive grasp of why an almost imperceptible change in the off-diagonal elements in a complex symmetric matrix causes it to change from being invertible to not being invertible. The diagonal elements are 1, and the sum of abs values of the off-diagonal elements in each row...

48. ### I How to interpret complex solutions to simple harmonic oscillator?

Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
49. ### LaTeX How to Represent Complex Fractions in LaTeX?

I know of two reasonable ways to represent a complex fraction: \dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) } ##\dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) }## and \dfrac{ ^a / _b }{ ^c / _d } ##\dfrac{ ^a / _b }{ ^c / _d }## What I am...
50. ### I Real ODE yields real solution through complex numbers

Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...