Fractional-linear function

A function of the type

$$ w = L ( z) = \frac{a _ {1} z _ {1} + \dots + a _ {n} z _ {n} + b }{c _ {1} z _ {1} + \dots + c _ {n} z _ {n} + d } , $$

where $ z = ( z _ {1} \dots z _ {n} ) $ are complex or real variables, $ a _ {j} $, $ b $, $ c _ {j} $, $ d $ are complex or real coefficients, and $ | c _ {1} | + \dots + | c _ {n} | + | d | > 0 $. If $ | c _ {1} | = \dots = | c _ {n} | = 0 $, the fractional-linear function is an integral-linear function; if the rank of the matrix

$$ A = \left \| \begin{array}{cccc} a _ {1} &\dots &a _ {n} & b \\ c _ {1} &\dots &c _ {n} & d \\ \end{array} \right \| $$

is equal to one, $ L ( z) $ is a constant. A proper fractional-linear function is obtained if $ | c _ {1} | + \dots + | c _ {n} | > 0 $ and if the rank of $ A $ is two; it assumed in what follows that these conditions have been met.

If $ n = 1 $ and $ a _ {1} = a $, $ c _ {1} = c $, $ z _ {1} = z $ are real, the graph of the fractional-linear function is an equilateral hyperbola with the asymptotes $ z = - d / c $ and $ w = a / c $. If $ n = 2 $ and $ a _ {1} $, $ a _ {2} $, $ b $, $ c _ {1} $, $ c _ {2} $, $ d $, $ z _ {1} $, $ z _ {2} $ are real, the graph of the fractional-linear function is hyperbolic paraboloid.

If $ n = 1 $, the fractional-linear function $ L ( z) $ is an analytic function of the complex variable $ z $ everywhere in the extended complex plane $ \overline{\mathbf C}\; $, except at the point $ z = - d / c $ at which $ L ( z) $ has a simple pole. If $ n \geq 1 $, the fractional-linear function $ L ( z) $ is a meromorphic function in the space $ \mathbf C ^ {n} $ of the complex variable $ z = ( z _ {1} \dots z _ {n} ) $, with the set

$$ \{ {z \in \mathbf C ^ {n} } : {c _ {1} z _ {1} + \dots + c _ {n} z _ {n} + d = 0 } \} $$

as its polar set.

See also Fractional-linear mapping.