📂Complex Anaylsis

Bilinear Transformation

Definition ¹

A mapping $f$ that is conformal in its domain is called as follows:

1. Translation $f(z) = z + \alpha$
2. Magnification: $f(z) = \rho z$
3. Rotation: $f(z) = e^{i \theta} z$
4. Inversion: $f(z) = {{1} \over {z}}$
5. Bilinear Transformation: $\displaystyle f(z) = {{ \alpha z + \beta } \over { \gamma z + \delta }}$

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• In translation, we have $\alpha \in \mathbb{C}$, and in magnification, we have $\rho \in \mathbb{R}^{ \ast }$.

Explanation

Among 1 to 4, the most distinct one is 4. Inversion, and therefore, only 1 to 3 combined are separately called Linear Transformation. No matter if a figure is moved, scaled, or rotated, i.e., undergoes a linear transformation, its shape itself is preserved; however, this is not the case with inversion.

On examining the bilinear transformation closely, it appears to be a combination of 1 to 4, firstly researched by Möbius, hence also known as the Möbius Transformation. When differentiated, it yields $\displaystyle f '(z) = {{ \alpha \delta - \beta \gamma } \over { ( \gamma z + \delta )^2 }}$, noting that to satisfy the condition of conformal mapping, it is necessary to adhere to $\alpha \delta \ne \beta \gamma$.