Finding roots and complex roots of a determinant

AI Thread Summary
The discussion focuses on finding the values of Ω from the equation involving determinants and complex roots. The initial equation simplifies to a fourth-degree polynomial, which the participants analyze for roots. Real roots are identified as ±√(2k/3m), but the challenge lies in determining the complex roots. The suggestion to use the identity a² - b² = (a + b)(a - b) is highlighted as a potential method for finding these roots. Ultimately, the complex roots are successfully identified using the notation provided by a participant.
Redwaves
Messages
134
Reaction score
7
Homework Statement
Finding roots and complex roots of a determinant
Relevant Equations
##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0##
I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0##

I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0##

I don't think this is the correct way.
I don't find a way to resolve a quadratic.
I have to factor some terms, but I don't see where.
 
Physics news on Phys.org
You have ##f(\Omega)^2=g(\Omega)^2##. Why don't you check all possibilities from here?
 
a^2 - b^2 = (a +b)(a - b) also holds for complex a and b...
 
I found the the reals roots which are ##\Omega = + -\sqrt{\frac{2k}{3m}}## using ##(-\Omega^2 +i\gamma\Omega + \frac{2k}{3m}))^2 = i^2\gamma^2\Omega^2##

However, I don't see how to get the complex roots.
To find the complex roots, should I replace ##i^2## by -1 ?
 
You have ##f(\Omega) =\pm g(\Omega)## or ##a=\pm b## with @pasmith 's notation.​

Now ##f(\Omega)## is quadratic, that gives you ##4## roots, two real and two complex.
 
Alright, Thanks!
I found it using Pasmith's notation.
Thanks guys!
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
Back
Top