How Do You Solve w=0.5(z+1/z) for Complex w?

[Macarenses]

How does one prove that for w≠+-1 , w a complex number, there are exactly 2 solutions to the equation w=0.5(z+1/z)? I'm at a total loss here. Could someone clue me in on this one?

 

[ramsey2879]

How does one prove that for w≠+-1 , w a complex number, there are exactly 2 solutions to the equation w=0.5(z+1/z)? I'm at a total loss here. Could someone clue me in on this one?

What if you multiplied each side by 2z?

 

[HallsofIvy]

The best way to prove there are two solutions is to find them! That is what ramsey2879 is suggesting.

 

[ramsey2879]

The best way to prove there are two solutions is to find them! That is what ramsey2879 is suggesting.

I do believe that the quadratic formula will work for equations with complex coefficients even if it is not clear (at least to me) how to find the square root of a complex number. Just write the solution out in the radical form simplified as far as posible.

 

[lostcauses10x]

Do a internet search: complex numbers solve square root

 

[TheBonobo4]

w=0.5(z+1/z)
w=0.5z+0.5/z
wz=0.5z^2+0.5
0.5z^2+0.5-wz=0

If you could substitute a value of either w or z you could work out the other using Quadratic Formula

x = (-b +- sqrt(b^2-4ac))/2a

Other than this, I can't see how you solve it since you have 2 unknowns. Either way you have a squared value for z so z has 2 solutions.

Sorry if I was no help, this is my first post here as I just joined today. :)