18.7: Fractional linear transformations
- Page ID
- 58351
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Watch video “Möbius transformations revealed” by Douglas Arnold and Jonathan Rogness. (It is available on YouTube.)
The complex plane \(\mathbb{C}\) extended by one ideal number \(\infty\) is called the extended complex plane. It is denoted by \(\hat{\mathbb{C}}\), so \(\hat{\mathbb{C}}=\mathbb{C} \cup\{\infty\}\)
A fractional linear transformation or Möbius transformation of \(\hat{\mathbb{C}}\) is a function of one complex variable \(z\) that can be written as
\(f(z) = \dfrac{a\cdot z + b}{c\cdot z + d},\)
where the coefficients \(a\), \(b\), \(c\), \(d\) are complex numbers satisfying \(a\cdot d - b\cdot c \not= 0\). (If \(a\cdot d - b\cdot c = 0\) the function defined above is a constant and is not considered to be a fractional linear transformation.)
In case \(c\not=0\), we assume that
\(f(-d/c) = \infty \quad \text{and} \quad f(\infty) = a/c;\)
and if \(c=0\) we assume
\(f(\infty) = \infty.\)