Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[ContinuedFraction] instead.

The cfrac command computes a continued fraction expansion of a number, rational polynomial, series, or other algebraic expression.

There are at least five forms for continued fraction expansions:

Regular continued fraction:

where, usually, $a_i$ and $b_i$ for i > 0 are integers or polynomials with integer coefficients. If $b_0\ne 0$ then it is called superdiagonal, otherwise it is called subdiagonal.

Simple continued fraction:

Simregular continued fraction:

Semisimple continued fraction:

where $\left|e_i\right|=1$, $\mathrm{sign}\left(b_i\right)=1$ for i = 1, 2, ...

Monic polynomial continued fraction:

where each $b_i$ is a monic polynomial for i = 1, 2, ...

By default, the cfrac command returns the fraction form of the continued fraction. The form used in the previous illustrations (1)-(5) is referred to as the list form.

The list form for a periodic continued fraction is $\left[\left[a_0\&comma;\mathrm{...}\&comma;a_m\right]\&comma;\left[b_1\&comma;\mathrm{...}\&comma;b_n\right]\right]$, where $0<m$, $1<n$, $a_0\in \mathbb{Z}$ and $a_1\&comma;\dots \&comma;a_n\&gt;0$ describe the preperiod and $b_1\&comma;\dots \&comma;b_n\&gt;0$ represent the period, i.e.,

If you specify the quotients option, the continued fraction is returned in list form.  For large continued fractions, this form prints more quickly than the fraction form.

If you specify the n option, at most n + 1 quotients of the continued fraction are computed.

If cfrac is passed a continued fraction, in fraction or list form, it computes the last convergent of the continued fraction. For a periodic continued fraction, it returns the corresponding quadratic surd.

The print routine print/CFRAC is used by the prettyprinter to format the fraction form on screen.

This function is part of the numtheory package, and so can be used in the form cfrac(..) only after performing the command with(numtheory). The function can always be accessed in the long form numtheory[cfrac](..).

The cfrac command computes a continued fraction expansion for three kinds of input:

Numeric

Rational polynomials

Series or algebraic objects

$\mathrm{with}\left(\mathrm{numtheory}\right)\&colon;$

$\mathrm{cfrac}\left(\mathrm{\pi }\&comma;6\right)$

$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\mathrm{...}}}}}}}$

$\mathrm{cfrac}\left(\mathrm{\pi }\&comma;100\&comma;'\mathrm{quotients}'\right)$

$\left[3\&comma;7\&comma;15\&comma;1\&comma;292\&comma;1\&comma;1\&comma;1\&comma;2\&comma;1\&comma;3\&comma;1\&comma;14\&comma;2\&comma;1\&comma;1\&comma;2\&comma;2\&comma;2\&comma;2\&comma;1\&comma;84\&comma;2\&comma;1\&comma;1\&comma;15\&comma;3\&comma;13\&comma;1\&comma;4\&comma;2\&comma;6\&comma;6\&comma;99\&comma;1\&comma;2\&comma;2\&comma;6\&comma;3\&comma;5\&comma;1\&comma;1\&comma;6\&comma;8\&comma;1\&comma;7\&comma;1\&comma;2\&comma;3\&comma;7\&comma;1\&comma;2\&comma;1\&comma;1\&comma;12\&comma;1\&comma;1\&comma;1\&comma;3\&comma;1\&comma;1\&comma;8\&comma;1\&comma;1\&comma;2\&comma;1\&comma;6\&comma;1\&comma;1\&comma;5\&comma;2\&comma;2\&comma;3\&comma;1\&comma;2\&comma;4\&comma;4\&comma;16\&comma;1\&comma;161\&comma;45\&comma;1\&comma;22\&comma;1\&comma;2\&comma;2\&comma;1\&comma;4\&comma;1\&comma;2\&comma;24\&comma;1\&comma;2\&comma;1\&comma;3\&comma;1\&comma;2\&comma;1\&comma;1\&comma;10\&comma;2\&comma;\mathrm{...}\right]$

$\mathrm{cfrac}\left(3^{\frac{1}{2}}\&comma;'\mathrm{periodic}'\right)$

$1+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\mathrm{...}}}}}$

$\mathrm{cfrac}\left(3^{\frac{1}{2}}\&comma;'\mathrm{periodic}'\&comma;'\mathrm{quotients}'\right)$

$\left[\left[1\right]\&comma;\left[1\&comma;2\right]\right]$

$\mathrm{cfrac}\left(\frac{35470}{99661}+\frac{315}{99661}\mathrm{I}\&comma;'\mathrm{centered}'\right)$

$\frac{\mathrm{I}}{3\mathrm{I}+\frac{1}{1+5\mathrm{I}-\frac{1}{2+\mathrm{I}+\frac{\mathrm{I}}{1+2\mathrm{I}-\frac{\mathrm{I}}{3+2\mathrm{I}-\frac{\mathrm{I}}{1+\mathrm{I}}}}}}}$

$f≔\mathrm{cfrac}\left(\frac{x^4+13x^2+50x+120}{5x^2+x+1}\right)$

$f≔\frac{x^2}{5}-\frac{x}{25}+\frac{321}{125}+\frac{1}{\frac{625x}{5934}-\frac{2810875}{11737452}+\frac{1}{\frac{23216680056x}{14242671875}+\frac{57431352636}{14242671875}}}$

$\mathrm{cfrac}\left(f\right)$

$\frac{x^4+13x^2+50x+120}{5x^2+x+1}$

$g≔\mathrm{cfrac}\left({\left(1+x\right)}^k\&comma;x\&comma;7\&comma;'\mathrm{subdiagonal}'\&comma;'\mathrm{simregular}'\right)$

$g≔\frac{1}{1-\frac{kx}{1+\frac{\left(k+1\right)x}{2\left(1-\frac{\left(k-1\right)x}{6\left(1+\frac{\left(k+2\right)x}{6\left(1-\frac{\left(k-2\right)x}{10\left(1+\frac{\left(k+3\right)x}{10\left(1+\mathrm{...}\right)}\right)}\right)}\right)}\right)}}}$

$\mathrm{op}\left(g\right)$

$\left[0\&comma;\left[1\&comma;1\right]\&comma;\left[-kx\&comma;1\right]\&comma;\left[\frac{\left(k+1\right)x}{2}\&comma;1\right]\&comma;\left[-\frac{\left(k-1\right)x}{6}\&comma;1\right]\&comma;\left[\frac{\left(k+2\right)x}{6}\&comma;1\right]\&comma;\left[-\frac{\left(k-2\right)x}{10}\&comma;1\right]\&comma;\left[\frac{\left(k+3\right)x}{10}\&comma;1\right]\&comma;\mathrm{...}\right]$

$\mathrm{cfrac}\left(\exp \left(x\right)\&comma;x=1\&comma;5\right)$

$\&ExponentialE;+\frac{x-1}{\frac{1}{\&ExponentialE;}+\frac{x-1}{-2\&ExponentialE;+\frac{x-1}{-\frac{3}{\&ExponentialE;}+\frac{x-1}{2\&ExponentialE;+\frac{x-1}{\frac{5}{\&ExponentialE;}+\mathrm{...}}}}}}$

The optional arguments con and den can be used in conjunction with the option centered as of Maple 16.

The argument ex_cf can be a periodic continued fraction as of Maple 16.

The a parameter was introduced in Maple 16.

The ex_cf parameter was updated in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.