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Continued fractions and linear fractional transformations

A linear fractional transformation (or a Möbius transformation) is a complex map of the form

f(z) = (a + b z)/(c + d z) ,(1)

where typeset structure are constants and typeset structure is a (generally complex) variable.  If typeset structure then typeset structure is constant. Therefore an additional restriction typeset structure removes this pathological case. typeset structureMoreover under this additional condition the linear fractional transformation is invertible, where the inverse map is also a linear fractional transformation, namely

f^(-1)(z) = (-a + c z)/(b - d w) .(2)

Transformation (1) has also a useful matrix representation

f |-> (b a) . d c(3)

This shows that composition of two non-constant linear fractional transformations is again a non-constant linear fractional transformation.

For more analytic properties of linear fractional transformations visit .

Consider the following sequence of linear fractional transformations

τ _ 0(z) = a _ 0 + z, τ _ 1(z) = b _ 1/(a _ 1 + z) , τ _ 2(z) = b _ 2/(a _ 2 + z) , τ _ 3(z) = b _ 3/(a _ 3 + z) , ... ,(4)

and their compositions

T _ n(z) = τ _ 0 o τ _ 1 o ... o τ _ n(z) = a _ 0 + b _ 1/(a _ 1 + b _ 2/(a _ 2 + b _ 3/(a _ 3 + _ (·.    a _ (n - 1) + b _ n/(a _ n + z)))))(5)

then

T _ n(0) = a _ 0 + b _ 1/(a _ 1 + b _ 2/(a _ 2 + b _ 3/(a _ 3 + _ (·.    a _ (n - 1) + b _ n/a _ n))))(6)

and moreover typeset structure.typeset structure

Using the fundamental recurrence formulas we get that

T _ n(z) = (P _ n + z P _ (n - 1))/(Q _ n + z Q _ (n - 1)) .(7)

As typeset structure is a continuous map (actually, it is a bijective conformal map of the extended complex plane typeset structure) and typeset structure, we get typeset structure maps every neighborhood of typeset structure into a neigbourhood of the value typeset structure of the continued fraction

a _ 0 + b _ 1/(a _ 1 + b _ 2/(a _ 2 + (   b _ 3)/(a _ 3 + _ ·.   )))(8)

provided (8) is convergent. The same conclusion is also true for the neighborhoods of typeset structure, that for all sufficiently large of typeset structure.

Cite this web-page as:

Štefan Porubský: Continued fractions and linear fractional transformations.

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