Linear Fractional Function

(also bilinear function), a function having the form

that is, the quotient of two linear functions. A linear fractional function is the simplest rational function. For ad - bc = 0, it reduces to a constant; if ad - bc = 0, but c = 0, then the bilinear function reduces to the linear function y = αx + β. Thus, we are interested only in the case where ad - bc = 0 and c - 0; the graph of the linear fractional function for real values of x is an equilateral hyperbola.

If x assumes arbitrary complex values (a, b, c, and d are fixed complex numbers), then a linear fractional function effects a one-to-one conformal mapping of the complex plane (supplemented by the point ∞) onto itself (this is the only analytic function having this property). The linear fractional function is also characterized by the fact that it maps lines and circles in the complex plane into lines and circles. Every conformal mapping of the interior of a circle onto itself can be realized by means of a linear fractional function. The cross ratio of four points

is an invariant of linear fractional functions. In other words, if a linear fractional function maps x₁ into y₁, x₂ into y₂, x₃ into y₃ and x₄ into y₄, then