Fractional-linear function
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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional-linear_function&oldid=13780