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# 18.7: Fractional linear transformations

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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

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