Hi guys. I need some help with I guess a relatively simple question. Could someone please show me with workings the following ?

If the modulus of z+(1/z)=2. What is z?

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- June 1st 2013, 03:57 AMYeah01Complex numbers
Hi guys. I need some help with I guess a relatively simple question. Could someone please show me with workings the following ?

If the modulus of z+(1/z)=2. What is z? - June 1st 2013, 04:12 AMchiroRe: Complex numbers
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)? - June 1st 2013, 04:17 AMYeah01Re: Complex numbers
Hi chiro,

Hope your having a good night thanks for your response. I've tried that still must be doing something wrong. :( - June 1st 2013, 04:26 AMchiroRe: Complex numbers
Hint: the modulus of z is SQRT(x^2 + y^2). If you are dealing with 1/z then the modulus is 1/SQRT(x^2 + y^2) since if z = r*e(i*theta) then 1/z = 1/r * e^(-i*theta) and 1/r is the modulus of 1/z.

- June 1st 2013, 04:38 AMYeah01Re: Complex numbers
Thanks chiro. Let me give it a go

- June 1st 2013, 04:43 AMYeah01Re: Complex numbers
Still no luck. Don't know what I'm doing wrong

- June 1st 2013, 05:07 AMYeah01Re: Complex numbers
Do u think u can help me out chiro?

- June 1st 2013, 03:32 PMYeah01Re: Complex numbers
Hi I got x = 1/4-y and y=1/4-x. What now? Sm I on the right track?

- June 1st 2013, 03:41 PMYeah01Re: Complex numbers
Hi I got x = 1/4-y and y=1/4-x. What now? Sm I on the right track?

- June 1st 2013, 04:03 PMPlatoRe: Complex numbers
I don't think at you are.

As you can see here, this problem has a rather nice answer.

But when I tried it, the algebra is really messy. From that link, I tried those solutions and they work. - June 1st 2013, 05:40 PMYeah01Re: Complex numbers
Thanks. What did yr working look like. The answer is good but really want work out how to do it.

- June 1st 2013, 06:40 PMPlatoRe: Complex numbers
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. - June 1st 2013, 07:35 PMGusbobRe: Complex numbers
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.

- June 1st 2013, 08:38 PMYeah01Re: Complex numbers
Wow. Thanks. Looking forward to seeing if someone can do the math.

- June 1st 2013, 08:58 PMGusbobRe: Complex numbers
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.