Möbius Transformation is a function of the form:

$$\begin{aligned} f(z) = \frac{az + b}{cz + d} \end{aligned}$$

where $a, b, c, d \in \mathbb{C}$ such that $ad - bc \not= 0$. (This looks surprisingly familiar to the determinant of a 2 by 2 invertible matrix !)

Interestingly,

• $f$ is a mapping from the extended complex plane $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ to $\hat{\mathbb{C}}$.
• $f$ is one-to-one and onto from $\hat{\mathbb{C}}$ to $\hat{\mathbb{C}}$ !
• $f$ does not uniquely determine the parameters $a, b, c, d$, for if we multiply each parameter by a constant $k \not= 0$, we obtain the same mapping.
• $\displaystyle f'(z) = \frac{ad - bc}{(cz + d)^2}$. Therefore, $f$ is a conformal mapping from $\hat{\mathbb{C}}$ to $\hat{\mathbb{C}}$.
• Möbius Transformation are the only conformal mappings from $\hat{\mathbb{C}}$ to $\hat{\mathbb{C}}$ !
• Every Möbius Transformation maps circles and lines to circles or lines !
• Given three distinct points $z_1,\, z_2,\, z_3 \in \hat{\mathbb{C}}$, there exists a unique Möbius Transformation $f$ such that $f(z_1) = 0,\, f(z_2) = 1,$ and $\, f(z_3) = \infty$ !

  Here it is:

  $$\begin{aligned} f(z) = \frac{z - z_1}{z - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1} \end{aligned}$$

Möbius Transformation is called Affine transformation when $c = 0$, and $d=1$. Interestingly,

• Affine transformations map $\infty$ to $\infty$ and therefore map $\mathbb{C}$ to $\mathbb{C}$.
• Hence Affine transformations are conformal mappings from $\mathbb{C}$ to $\mathbb{C}$.
• Affine transformations are the only conformal mappings from $\mathbb{C}$ to $\mathbb{C}$ !
• When $b = 0,\, f(z) = az$ which is a rotation and dilation.
• When $a = 1,\, f(z) = z + b$ which is a translation.
• When $\displaystyle a = 0, b = 1, c = 1, d = 0, f(z) = \frac{1}{z}$ which is an inversion.
  • $f$ interchanges outside and inside of the unit circle
  • a circle centered at $0$ is clearly mapped to a circle, centered at $0$, of reciprocal radius.

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 8:44 AM