Find any fixed points for the following mapping:

[andrey21]

Find any fixed points for following mapping

f(z) = z² - z + 1

Map onto the same point gives:

z = z² - z + 1
0 = z² - 2z + 1

Therefore z = 1 and z = 1

 

[tiny-tim]

hi andrey21!

that looks ok …

what is worrying you about that? ​

 

[andrey21]

Just looking for confirmation really , I do have a similar question which I am having trouble with:

f(z) = 2i/z

Not sure how to approach this one:

z = 2i/z
z² = 2i

This is as far as I have gotten:

 

[tiny-tim]

z² = 2i

every (non-zero) complex number has a unique modulus (positive real number) and argument (in [0,2π)) …

what happens to that modulus and argument when you square the number?

 

[andrey21]

Ok so if I square the number I get:

2i² = -4

Am I on the write track?

 

[tiny-tim]

no, i mean square z (and see if you get the modulus and argument of 2i)

 

[andrey21]

Ok I am a little confused, you are saying I should do the following:

=(0+2i/z)²??

If so would this give -4/z²??

 

[tiny-tim]

no, I'm saying you should square (modulus)e^i(argument) to see if it gives 2i

 

[andrey21]

I see well the modulus is:

=SQRT(x²+y²)
=SQRT(0+4)

=2

correct? I can't recall how to obtain the argument.

 

[tiny-tim]

the argument is just a silly old-fashioned name for the angle (with the positive real axis measured as angle 0)

so the argument (angle) of x + iy has tangent y/x

the argument of 2i = … ?​

 

[andrey21]

Is it =0??

 

[tiny-tim]

tan0 = 1/0 ?