A function of the type
where are complex or real variables, , , , are complex or real coefficients, and . If , the fractional-linear function is an integral-linear function; if the rank of the matrix
is equal to one, is a constant. A proper fractional-linear function is obtained if and if the rank of is two; it assumed in what follows that these conditions have been met.
If and , , are real, the graph of the fractional-linear function is an equilateral hyperbola with the asymptotes and . If and , , , , , , , are real, the graph of the fractional-linear function is hyperbolic paraboloid.
If , the fractional-linear function is an analytic function of the complex variable everywhere in the extended complex plane , except at the point at which has a simple pole. If , the fractional-linear function is a meromorphic function in the space of the complex variable , with the set
as its polar set.
See also Fractional-linear mapping.
Fractional-linear function. E.P. DolzhenkoE.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fractional-linear_function&oldid=13780