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

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tiny-tim

Homework Helper
hi andrey21!

that looks ok …

what is worrying you about that?

andrey21

Just looking for confirmation really , I do have a similar question which im 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

Homework Helper
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

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

andrey21

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

=(0+2i/z)2??

If so would this give -4/z2??

tiny-tim

Homework Helper
no, i'm saying you should square (modulus)ei(argument) to see if it gives 2i

andrey21

I see well the modulus is:

=SQRT(x2+y2)
=SQRT(0+4)

=2

correct? I cant recall how to obtain the argument.

tiny-tim

Homework Helper
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 = … ?​

Is it =0??

Homework Helper
tan0 = 1/0 ?

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