Definition:Continued Fraction Expansion

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Definition

Continued fraction expansion of a real number

Irrational Number

Let $x$ be an irrational number.


The continued fraction expansion of $x$ is the simple continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as:

$\alpha_n = \displaystyle \begin{cases} x & : n = 0 \\ \dfrac 1 {\{\alpha_{n-1}\}} & : n \geq 1 \end{cases}$

where:

$\lfloor \cdot \rfloor$ is the floor function
$\{ \cdot \}$ is the fractional part function.


Rational Number

Let $x$ be a rational number.


The continued fraction expansion of $x$ is found using the Euclidean Algorithm.


Continued fraction expansion of a Laurent series

Let $k$ be a field.

Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$.


Irrational Laurent series

Let $f \in k((t^{-1}))$ be an irrational formal Laurent series.


The continued fraction expansion of $f$ is the infinite continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as:

$\alpha_n = \displaystyle \begin{cases} f & : n = 0 \\ \dfrac 1 {f - \lfloor f \rfloor} & : n \geq 1 \end{cases}$

where $\lfloor \cdot \rfloor$ denotes the polynomial part.


Rational Laurent series


By definition, a continued fraction equals its sequence of partial quotients.

Therefore, to reduce the cumbersome nature of its representation, the continued fraction in the definitions are usually written as:

$\left[{a_0, a_1, a_2, \ldots, a_n}\right]$

for the finite case, and:

$\left[{a_0, a_1, a_2, \ldots}\right]$

for the infinite case.


Such an expression is known as the continued fraction expansion of the continued fraction, especially in the case of the infinite version.

For example:

$\left[{1, 2, 3}\right] = 1 + \cfrac 1 {2 + \cfrac 1 3}$


Also see