Hey Yeah01.
Hint: Can you calculate an expression for the modulus in terms of x and y (for x + iy) or in terms of r and theta (polar co-ordinates)?
(In other words, if you have z + 1/z, what is the modulus in terms of x and y if z = x + iy)?
Hey Yeah01.
Hint: Can you calculate an expression for the modulus in terms of x and y (for x + iy) or in terms of r and theta (polar co-ordinates)?
(In other words, if you have z + 1/z, what is the modulus in terms of x and y if z = x + iy)?
Well I really have absolutely no idea how to work it out algebraically.
I taught complex variables on and off for over thirty years. But I have never seen a more difficult problem in this category of problems than this one is.
Clearly are solutions simply by inspection.
But how one gets the other eight solution remains a puzzle to me.
There may well be a very clever trick that I have not thought of.
I believe the most intuitive way to do this is to look at the pre image of the Joukovski's transformation . This is a conformal map, and there are some nice ways you can deal with this. I also believe it can be cranked out algebraically in an acceptably nice way. I'm working on that now.
Let . Then the condition in the question is equivalent to saying . If , you'll recover the trivial answers
Exercise: If and , show that is or . That is, is purely imaginary.
HINT: Writing , you should find that
Fun facts: and
Hence for some real number a, with some restrictions on .
Now you need some bounds on since you had the restriction ,
I didn't have time to finish calculating the bounds, but note that and gives the easier solutions in Plato's wolfram alpha link. It comes out by solving .
Good luck with that. I'll work on the bounds later if I have time.