Find a linear fractional transformation that carries circle to a line.

In summary, the conversation discusses the transformation L defined by the equation |z| = 1 and Re(1 + w) = 0. The conversation also mentions using three unique points to determine the transformation L and finding the transformation in a simple form.
  • #1
StumpedPupil
11
0

Homework Statement



Define L: |z| = 1 -----> Re( (1 + w)) = 0. Find L.

Homework Equations



A transformation is defined by three unique points by T(z) = (z-z1)(z2-z3) / (z-z3)(z2-z1). If we have two transformations T and S, and we want T = S for three distinct points, then we have the transformation L by the transformation S^(-1)[T(z)].

The Attempt at a Solution



I chose the points on the circle 1, i, and -1 to go to the points infinity, 0 and 1+i respectively. My calculations gave me L(z) = (1+i)((z-i) - (z+1))(infinity) / ((infinity)(z-1)i - (z+1)(1+i)). The book gives me u(1-i)(z+1)/(z-1) where u is any real number.

What should I do to get this simple form (aka the right answer). Thank you.

 
Physics news on Phys.org
  • #2
Why are you mapping points to 0 and 1 + i? Doesn't Re(1 + w) = 0 represent a vertical line?

Cancel out the terms containing infinity.
 
  • #3
I'm sorry, that is a typo. I meant to write Re((1 + i)w) = 0. This is the line y = x. I choose two points on this line, and a point at infinity.
 

FAQ: Find a linear fractional transformation that carries circle to a line.

1. What is a linear fractional transformation?

A linear fractional transformation is a mathematical function that maps points from one complex plane to another, using a ratio of linear polynomials.

2. How does a linear fractional transformation carry a circle to a line?

A linear fractional transformation can be used to transform points on a circle in the complex plane to points on a line in the complex plane by mapping the center of the circle to infinity and the circumference of the circle to a straight line.

3. Can any circle be transformed into a line using a linear fractional transformation?

Yes, any circle in the complex plane can be transformed into a line using a linear fractional transformation. However, the specific transformation used will depend on the circle's center and radius.

4. How can I find the linear fractional transformation that will carry a specific circle to a line?

To find the linear fractional transformation, you will need to know the center and radius of the circle, as well as the desired line that you want the circle to be transformed into. The transformation can be found by solving a system of equations using the known points on the circle and the corresponding points on the line.

5. Are there any other applications of linear fractional transformations?

Yes, linear fractional transformations have many applications in mathematics and physics. They can be used to solve complex equations, transform geometric shapes, and even model physical phenomena such as fluid flow and electromagnetic fields.

Similar threads

Back
Top