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Mathematics LibreTexts

18.7: Fractional linear transformations

  • Page ID
    58351
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    Exercise \(\PageIndex{1}\)

    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

    \(f(\infty) = \infty.\)