Linear fractional transformation

In summary, the conversation discusses the use of Schwarz's Reflection Principle and the Cayley transform in determining conformal mappings from the upper half plane to the unit disk. The speaker is having difficulty understanding how the principle is applied and why there is no non-constant function that is real-valued on the unit circle. Links are provided for further information.
  • #1
arthurhenry
43
0
The following comes from the complex analysis text by Joseph Bak:



He is trying to determine all conformal mappings f of upper half plane H onto the unit disk.

"Let us first assume that f is an LFT and f(a)=0 for Im(a)>0.
Then, since the real axis is mapped into the unit circle, it follows by Schwarz Principle of reflection that f(a*)=Infinity, so that...

Here a* denotes the conjugate of a.

I am having difficulty following how the Schwarz Reflection Principle is applied...and we get that f(a*)=Infinity.

thank you.

Also, Isn't true that the inverse Caylet Transform map the unit circle into real axis.If so, why "there is no non-constant analytic function in the unit disk that is real valued on the unit circle" ?
Thank you
 
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  • #2

1. What is a linear fractional transformation?

A linear fractional transformation is a mathematical function that maps a complex number to another complex number. It is also known as Möbius transformation or a bilinear transformation. It can be represented in the form of (az+b)/(cz+d), where a, b, c and d are complex numbers.

2. What is the significance of linear fractional transformation?

Linear fractional transformation is important in complex analysis and geometry. It is used to map geometric shapes in the complex plane, such as circles and lines, to other shapes. It is also used in solving problems related to conformal mapping, which is used in engineering and physics.

3. What are the properties of linear fractional transformation?

The properties of linear fractional transformation include preserving lines and circles, preserving angles, and preserving the cross-ratio. It is also a one-to-one mapping, meaning that each point in the domain has a unique image in the range.

4. How is linear fractional transformation related to matrices?

A linear fractional transformation can be represented by a 2x2 matrix. This matrix is called the Möbius matrix and is defined as [a b;c d]. The linear fractional transformation can be obtained by dividing the entries of the matrix by their determinant. This relationship is useful in solving problems involving linear fractional transformation.

5. Can linear fractional transformation be extended to higher dimensions?

Yes, linear fractional transformation can be extended to higher dimensions. In 3D space, it can be represented by a 3x3 matrix. In general, linear fractional transformation can be extended to n dimensions by using an (n+1)x(n+1) matrix. However, its properties such as preserving lines and circles may not hold in higher dimensions.

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