# Show the pre-image of a vertical line Re(w)=c under f

• Raziel2701
In summary, the pre-image of a vertical line Re(w)=c under the function f is a circle centered on the line Re(z) = -1 and tangent to the real axis Im(z) = 0. This circle intersects the unit circle |z| = 1 at two points, forming a 90 degree angle. In order to find the real part of z, we can use the formula (u+ \overline{u})/2, where u is the complex number in question.
Raziel2701
Show the pre-image of a vertical line Re(w)=c under f...

## Homework Statement

a) Show that the pre-image of a vertical line Re(w)=c under f is a circle centered on the line Re(z) = -1 and tangent to the real axis Im(z) = 0.

b) How does such a circle intersect the circle |z| =1?

Here's f by the way: $$i\frac{1-z}{1+z}$$

## The Attempt at a Solution

So I took f and set it equal to c:

$$i\frac{1-z}{1+z} =c$$

Solving for z, I get that the Re(z)

$$Re(z) = \frac{-c^2 +1}{c^2 +1}$$

So that was me not knowing what I'm doing and just messing with the given information. I then took the limit as z approaches infinity and I get -1, so I again, not knowing if I'm doing this right simply declared that Re(z) is now equal to -1 as I'm supposed to prove.

So am I doing this right?

For part b, the circles intersect twice at 90 degrees right?

Last edited:

f(z) is NOT equal to c, its real part is. The real part of any complex number, u, is equal to $(u+ \overline{u})/2$. Find that for this function
$$\overline{f}(z)= f(\overline{z})$$
and set it equal to 0.

## 1. What is a pre-image?

A pre-image is the original figure or shape before any transformation or function is applied to it.

## 2. What is a vertical line?

A vertical line is a straight line that goes straight up and down, and has a slope of undefined or zero.

## 3. What does "Re(w)=c" mean?

"Re(w)=c" is a mathematical notation that represents the real part of a complex number "w" being equal to a constant "c". In other words, it is a vertical line in the complex plane at a specific real value "c".

## 4. What is f in this context?

In this context, f represents a function or transformation that is being applied to the pre-image of the vertical line. It could be any mathematical operation or rule.

## 5. How do you show the pre-image of a vertical line under a function?

To show the pre-image of a vertical line under a function, you would need to plot the original vertical line on a graph, then apply the function to each point on the line to determine the new coordinates. The resulting points can then be plotted to show the pre-image of the vertical line under the given function.

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