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.