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