Find any fixed points for the following mapping:

In summary, The fixed points for the mapping f(z) = z2 - z + 1 are z = 1 and z = 1hi, and the fixed points for the mapping f(z) = 2i/z can be found by squaring the modulus and argument of 2i, which are 2 and 0, respectively. The argument of a complex number is defined as the angle with the positive real axis, measured from 0.
  • #1
andrey21
476
0
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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  • #2
hi andrey21! :smile:

that looks ok …

what is worrying you about that? :wink:
 
  • #3
Just looking for confirmation really :smile:, 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:
 
  • #4
andrey21 said:
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? :wink:
 
  • #5
Ok so if I square the number I get:

2i2 = -4

Am I on the write track?
 
  • #6
no, i mean square z (and see if you get the modulus and argument of 2i)
 
  • #7
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??
 
  • #8
no, I'm saying you should square (modulus)ei(argument) to see if it gives 2i
 
  • #9
I see well the modulus is:

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

=2

correct? I can't recall how to obtain the argument.
 
  • #10
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 :wink:

the argument of 2i = … ?​
 
  • #11
Is it =0??
 
  • #12
tan0 = 1/0 ? :confused:
 

1. What is a fixed point in a mapping?

A fixed point in a mapping is a value that does not change after being transformed by the mapping. In other words, the output of the mapping is equal to its input at a fixed point.

2. How do you find fixed points in a mapping?

To find fixed points in a mapping, set the output of the mapping equal to its input and solve for the variable. The solution(s) will be the fixed point(s) of the mapping.

3. Why is it important to find fixed points in a mapping?

Fixed points in a mapping can provide valuable information about the behavior and stability of the mapping. They can also be used to simplify complex equations and systems.

4. Can a mapping have more than one fixed point?

Yes, a mapping can have more than one fixed point. In fact, some mappings may have an infinite number of fixed points.

5. How do fixed points relate to the graph of a mapping?

The fixed points of a mapping can be represented by the points where the graph of the mapping intersects with the line y=x. This means that at a fixed point, the input and output of the mapping are equal.

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