Generalized Continued Fraction - Linear Fractional Transformations

Linear Fractional Transformations

A linear fractional transformation (LFT) is a complex function of the form

where z is a complex variable, and a, b, c, d are arbitrary complex constants. An additional restriction – that ad ≠ bc – is customarily imposed, to rule out the cases in which w = f(z) is a constant. The linear fractional transformation, also known as a Möbius transformation, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions.

• If d ≠ 0 the LFT has one or two fixed points. This can be seen by considering the equation

which is clearly a quadratic equation in z. The roots of this equation are the fixed points of f(z). If the discriminant (c − b)2 + 4ad is zero the LFT fixes a single point; otherwise it has two fixed points.

• If ad ≠ bc the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function

such that f(g(z)) = g(f(z)) = z for every point z in the extended complex plane, and both f and g preserve angles and shapes at vanishingly small scales. From the form of z = g(w) we see that g is also an LFT.

• The composition of two different LFTs for which ad ≠ bc is itself an LFT for which ad ≠ bc. In other words, the set of all LFTs for which ad ≠ bc is closed under composition of functions. The collection of all such LFTs – together with the "group operation" composition of functions – is known as the automorphism group of the extended complex plane.

• If b = 0 the LFT reduces to

which is a very simple meromorphic function of z with one simple pole (at −c/d) and a residue equal to a/d. (See also Laurent series.)