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FP2 complex numbers problem Watch

1. The z-plane is mapped onto the w-plane by the transformation w=z+1/z . Given that z lies on the circle |z|=1, show that w lies on an interval of the real axis. Identify this interval precisely.

I've done:
w=(z^2 +1 )/z
|w|=|z^2 +1|/|z|
|w|=|z^2 +1|

Not sure what to do next
2. (Original post by bobbricks)
The z-plane is mapped onto the w-plane by the transformation w=z+1/z . Given that z lies on the circle |z|=1, show that w lies on an interval of the real axis. Identify this interval precisely.

I've done:
w=(z^2 +1 )/z
|w|=|z^2 +1|/|z|
|w|=|z^2 +1|

Not sure what to do next
I'd take another approach.

3. (Original post by BuryMathsTutor)
I'd take another approach.

So 1/z =x-iy as |z|=x^2 + y^2 =1 ?

So |w|=x-iy+1 ...?
4. (Original post by bobbricks)
So 1/z =x-iy as |z|=x^2 + y^2 =1 ?

So |w|=x-iy+1 ...?

Yes, 1/z=x-iy.

So what is ?
5. (Original post by BuryMathsTutor)
Yes, 1/z=x-iy.

So what is ?
Z+1/Z = 2x

so |w|=2|x| ...?
6. (Original post by bobbricks)
Z+1/Z = 2x

so |w|=2|x| ...?

and .
7. (Original post by BuryMathsTutor)
and .
How do you know that?
8. (Original post by bobbricks)
How do you know that?
Your question said: "Given that z lies on the circle |z|=1".
9. (Original post by bobbricks)
The z-plane is mapped onto the w-plane by the transformation w=z+1/z . Given that z lies on the circle |z|=1, show that w lies on an interval of the real axis. Identify this interval precisely.

I've done:
w=(z^2 +1 )/z
|w|=|z^2 +1|/|z|
|w|=|z^2 +1|

Not sure what to do next

although this particular question can be done in Cartesian
this type of problem is solved by the substitution z = e, which represent a unit circle

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