We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
120
1
avatar

Let \(w\) and \(z\) be complex numbers such that \(w = z + \frac{1}{z} = z^2 + \frac{1}{z^2}\). Find all possible values of \(w\).

 Apr 5, 2019
 #1
avatar+6042 
+1

\(z+\dfrac 1 z = z^2 + \dfrac{1}{z^2}\\ z^4-z^3-z+1=0\)

 

\(\text{A quick eyeball solution is }z=1\\ \text{So let's divide through by }(z-1)\\ z^3-1 = 0\\ \text{This also clearly has a solution at }z=1 \text{ so divide again}\\ z^2+z+1 = 0\\ \text{This is just quadratic so we can apply the quadratic formula}\\ z = \dfrac{-1\pm \sqrt{1-4}}{2} = -\dfrac 1 2 \pm i\sqrt{3}\\ \text{So in total we have}\\ z = 1, ~z= -\dfrac 1 2 + \dfrac{i\sqrt{3}}{2}, ~z = -\dfrac 1 2 - \dfrac{i\sqrt{3}}{2}\)

 

\(w = 1 + \dfrac 1 1 = 2\\ -\dfrac 1 2 \pm i \sqrt{3} = e^{\pm i\frac{2\pi}{3}}\\ e^{\pm i\frac{2\pi}{3}}+e^{\mp i\frac{2\pi}{3}} = 2\cos\left(\dfrac{2\pi}{3}\right) = -1 \)

.
 Apr 5, 2019
edited by Rom  Apr 5, 2019
edited by Rom  Apr 6, 2019

5 Online Users