Generalized continued fraction

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.

A generalized continued fraction is an expression of the form

x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

x_{0}={\frac {A_{0}}{B_{0}}}=b_{0},\qquad x_{1}={\frac {A_{1}}{B_{1}}}={\frac {b_{1}b_{0}+a_{1}}{b_{1}}},\qquad x_{2}={\frac {A_{2}}{B_{2}}}={\frac {b_{2}(b_{1}b_{0}+a_{1})+a_{2}b_{0}}{b_{2}b_{1}+a_{2}}},\qquad \cdots \,

and in general^[1]

A_{n}=b_{n}A_{n-1}+a_{n}A_{n-2},\qquad B_{n}=b_{n}B_{n-1}+a_{n}B_{n-2},\,

where A_n is the numerator and B_n is the denominator, called continuants,^[2]^[3] of the nth convergent.

If the sequence of convergents {x_n} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators B_n.

History

The story of continued fractions begins with the Euclidean algorithm,^[4] a procedure for finding the greatest common divisor of two natural numbers m and n. That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder repeatedly.

Nearly two thousand years passed before Rafael Bombelli^[5] devised a technique for approximating the roots of quadratic equations with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi introduced the first formal notation^[6] for the generalized continued fraction. Cataldi represented a continued fraction as

a_{0}.\,

{n_{1} \over d_{1}.}

{n_{2} \over d_{2}.}

{n_{3} \over d_{3}},

with the dots indicating where the next fraction goes, and each & representing a modern plus sign.

Late in the seventeenth century John Wallis^[7] introduced the term "continued fraction" into mathematical literature. New techniques for mathematical analysis (Newton's and Leibniz's calculus) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use.

In 1748 Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series.^[8] Euler's continued fraction formula is still the basis of many modern proofs of convergence of continued fractions.

In 1761, Johann Heinrich Lambert gave the first proof that $π$ is irrational, by using the following continued fraction for $tan x$ :^[9]

\tan(x)={\cfrac {x}{1+{\cfrac {-x^{2}}{3+{\cfrac {-x^{2}}{5+{\cfrac {-x^{2}}{7+{}\ddots }}}}}}}}

Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years.^[10] Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic string of length p - 1.

In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions.^[11] They can be used to express many elementary functions and some more advanced functions (such as the Bessel functions), as continued fractions that are rapidly convergent almost everywhere in the complex plane.

Notation

The long continued fraction expression displayed in the introduction is probably the most intuitive form for the reader. Unfortunately, it takes up a lot of space in a book (and it's not easy for the typesetter, either). So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this:

x=b_{0}+{\frac {a_{1}}{b_{1}+}}\,{\frac {a_{2}}{b_{2}+}}\,{\frac {a_{3}}{b_{3}+}}\cdots

Pringsheim wrote a generalized continued fraction this way:

x=b_{0}+{\frac {a_{1}\mid }{\mid b_{1}}}+{\frac {a_{2}\mid }{\mid b_{2}}}+{\frac {a_{3}\mid }{\mid b_{3}}}+\cdots \,

Carl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation:

x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}.\,

Here the "K" stands for Kettenbruch, the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.

Some elementary considerations

Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions.

Partial numerators and denominators

If one of the partial numerators a_n+1 is zero, the infinite continued fraction

b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,

is really just a finite continued fraction with n fractional terms, and therefore a rational function of the first n a_i's and the first (n + 1) b_i's. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that none of the a_i = 0. There is no need to place this restriction on the partial denominators b_i.

The determinant formula

When the nth convergent of a continued fraction

x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,

is expressed as a simple fraction x_n = A_n/B_n we can use the determinant formula

$A_{n-1}B_{n}-A_{n}B_{n-1}=(-1)^{n}a_{1}a_{2}\cdots a_{n}=\Pi _{i=1}^{n}(-a_{i})$

(1)

to relate the numerators and denominators of successive convergents x_n and x_n-1 to one another. The proof for this can be easily seen by induction.

Base Case

It is trivially true.

Inductive Step

Assume the (1) holds for $n-1$ . Then we need to see the same relation holding true for $n$ . Substituting the value of $A_{n}$ and $B_{n}$ in 1 we obtain:

${\begin{aligned}&=b_{n}A_{n-1}B_{n-1}+a_{n}A_{n-1}B_{n-2}-b_{n}A_{n-1}B_{n-1}-a_{n}A_{n-2}B_{n-1}\\&=a_{n}(A_{n-1}B_{n-2}-A_{n-2}B_{n-1})\end{aligned}}$

which is true because of our induction hypothesis.

$A_{n-1}B_{n}-A_{n}B_{n-1}=(-1)^{n}a_{1}a_{2}\cdots a_{n}=\Pi _{i=1}^{n}(-a_{i})\,$

Specifically, if neither B_n nor B_n-1 is zero we can express the difference between the n-1st and nth (n > 0) convergents like this:

x_{n-1}-x_{n}={\frac {A_{n-1}}{B_{n-1}}}-{\frac {A_{n}}{B_{n}}}=(-1)^{n}{\frac {a_{1}a_{2}\cdots a_{n}}{B_{n}B_{n-1}}}={\frac {\Pi _{i=1}^{n}(-a_{i})}{B_{n}B_{n-1}}}.\,

The equivalence transformation

If {c_i} = {c₁, c₂, c₃, ...} is any infinite sequence of non-zero complex numbers we can prove, by induction, that

b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}=b_{0}+{\cfrac {c_{1}a_{1}}{c_{1}b_{1}+{\cfrac {c_{1}c_{2}a_{2}}{c_{2}b_{2}+{\cfrac {c_{2}c_{3}a_{3}}{c_{3}b_{3}+{\cfrac {c_{3}c_{4}a_{4}}{c_{4}b_{4}+\ddots \,}}}}}}}}

where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right.

The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the a_i are zero a sequence {c_i} can be chosen to make each partial numerator a 1:

b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{c_{i}b_{i}}}\,

where c₁ = 1/a₁, c₂ = a₁/a₂, c₃ = a₂/(a₁a₃), and in general c_n+1 = 1/(a_n+1c_n).

Second, if none of the partial denominators b_i are zero we can use a similar procedure to choose another sequence {d_i} to make each partial denominator a 1:

b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {d_{i}a_{i}}{1}}\,

where d₁ = 1/b₁ and otherwise d_n+1 = 1/(b_nb_n+1).

These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.

Simple convergence concepts

It has already been noted that the continued fraction

x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,

converges if the sequence of convergents {x_n} tends to a finite limit.

The notion of absolute convergence plays a central role in the theory of infinite series. No corresponding notion exists in the analytic theory of continued fractions – in other words, mathematicians do not speak of an absolutely convergent continued fraction. Sometimes the notion of absolute convergence does enter the discussion, however, especially in the study of the convergence problem. For instance, a particular continued fraction

x={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{b_{i}}}\,

diverges by oscillation if the series b₁ + b₂ + b₃ + ... is absolutely convergent.^[12]

Sometimes the partial numerators and partial denominators of a continued fraction are expressed as functions of a complex variable z. For example, a relatively simple function^[13] might be defined as

f(z)={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{z}}.\,

For a continued fraction like this one the notion of uniform convergence arises quite naturally. A continued fraction of one or more complex variables is uniformly convergent in an open neighborhood Ω if the fraction's convergents converge uniformly at every point in Ω. Or, in gory detail: if, for every ε > 0 an integer M can be found such that the absolute value of the difference

f(z)-f_{n}(z)={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}(z)}{b_{i}(z)}}-{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}(z)}{b_{i}(z)}}\,

is less than ε for every point z in an open neighborhood Ω whenever n > M, the continued fraction defining f(z) is uniformly convergent on Ω. (Here f_n(z) denotes the nth convergent of the continued fraction, evaluated at the point z inside Ω, and f(z) is the value of the infinite continued fraction at the point z.)

The Śleszyński–Pringsheim theorem provides a sufficient condition for convergence.

Even and odd convergents

It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points p and q, then the sequence {x₀, x₂, x₄, ...} must converge to one of these, and {x₁, x₃, x₅, ...} must converge to the other. In such a situation it may be convenient to express the original continued fraction as two different continued fractions, one of them converging to p, and the other converging to q.

The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if

x={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{1}}\,

is a continued fraction, then the even part x_even and the odd part x_odd are given by

x_{\mathrm {even} }={\cfrac {a_{1}}{1+a_{2}-{\cfrac {a_{2}a_{3}}{1+a_{3}+a_{4}-{\cfrac {a_{4}a_{5}}{1+a_{5}+a_{6}-{\cfrac {a_{6}a_{7}}{1+a_{7}+a_{8}-\ddots }}}}}}}}\,

and

x_{\mathrm {odd} }=a_{1}-{\cfrac {a_{1}a_{2}}{1+a_{2}+a_{3}-{\cfrac {a_{3}a_{4}}{1+a_{4}+a_{5}-{\cfrac {a_{5}a_{6}}{1+a_{6}+a_{7}-{\cfrac {a_{7}a_{8}}{1+a_{8}+a_{9}-\ddots }}}}}}}}\,

respectively. More precisely, if the successive convergents of the continued fraction x are {x₁, x₂, x₃, ...}, then the successive convergents of x_even as written above are {x₂, x₄, x₆, ...}, and the successive convergents of x_odd are {x₁, x₃, x₅, ...}.^[14]

Conditions for irrationality

If $a_{1},a_{2},{\text{ . . .}}$ and $b_{1},b_{2},{\text{ . . .}}$ are positive integers with $a_{k}$ ≤ $b_{k}$ for all sufficiently large $k$ , then

x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,

converges to an irrational limit.^[15]

Fundamental recurrence formulas

The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas:

{\begin{aligned}A_{-1}&=1&B_{-1}&=0\\A_{0}&=b_{0}&B_{0}&=1\\A_{n+1}&=b_{n+1}A_{n}+a_{n+1}A_{n-1}&B_{n+1}&=b_{n+1}B_{n}+a_{n+1}B_{n-1}\,\end{aligned}}

The continued fraction's successive convergents are then given by

x_{n}={\frac {A_{n}}{B_{n}}}.\,

These recurrence relations are due to John Wallis (1616-1703) and Leonhard Euler (1707-1783).^[16]

As an example, consider the regular continued fraction in canonical form that represents the golden ratio φ:

x=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots \,}}}}}}}}

Applying the fundamental recurrence formulas we find that the successive numerators A_n are {1, 2, 3, 5, 8, 13, ...} and the successive denominators B_n are {1, 1, 2, 3, 5, 8, ...}, the Fibonacci numbers. Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly.

Linear fractional transformations

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

w=f(z)={\frac {a+bz}{c+dz}},\,

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

f(z)=z\Rightarrow dz^{2}+cz=a+bz\,

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

z=g(w)={\frac {-a+cw}{b-dw}}\,

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

w=f(z)={\frac {a}{c+dz}},\,

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.)

The continued fraction as a composition of LFTs

Consider a sequence of simple linear fractional transformations

\tau _{0}(z)=b_{0}+z,\quad \tau _{1}(z)={\frac {a_{1}}{b_{1}+z}},\quad \tau _{2}(z)={\frac {a_{2}}{b_{2}+z}},\quad \tau _{3}(z)={\frac {a_{3}}{b_{3}+z}},\quad \cdots \,

Here we use the Greek letter τ (tau) to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol Τ_n to represent the composition of n+1 little τs – that is,

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)=\tau _{0}\circ \tau _{1}(z)=\tau _{0}(\tau _{1}(z)),\quad {\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)=\tau _{0}(\tau _{1}(\tau _{2}(z))),\,

and so forth. By direct substitution from the first set of expressions into the second we see that

{\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)&=\tau _{0}\circ \tau _{1}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+z}}\\{\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)&=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+z}}}}\,\end{aligned}}

and, in general,

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)=\tau _{0}\circ \tau _{1}\circ \tau _{2}\circ \cdots \circ \tau _{n}(z)=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,

where the last partial denominator in the finite continued fraction K is understood to be b_n + z. And, since b_n + 0 = b_n, the image of the point z = 0 under the iterated LFT Τ_n is indeed the value of the finite continued fraction with n partial numerators:

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}.\,

A geometric interpretation

Defining a finite continued fraction as the image of a point under the iterated linear functional transformation Τ_n(z) leads to an intuitively appealing geometric interpretation of infinite continued fractions.

The relationship

x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}={\frac {A_{n}}{B_{n}}}={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )\,

can be understood by rewriting Τ_n(z) and Τ_n+1(z) in terms of the fundamental recurrence formulas:

{\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {(b_{n}+z)A_{n-1}+a_{n}A_{n-2}}{(b_{n}+z)B_{n-1}+a_{n}B_{n-2}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}};\\{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {(b_{n+1}+z)A_{n}+a_{n+1}A_{n-1}}{(b_{n+1}+z)B_{n}+a_{n+1}B_{n-1}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {zA_{n}+A_{n+1}}{zB_{n}+B_{n+1}}}.\,\end{aligned}}

In the first of these equations the ratio tends toward A_n/B_n as z tends toward zero. In the second, the ratio tends toward A_n/B_n as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents A_n/B_n are eventually arbitrarily close together. Since the linear fractional transformation Τ_n(z) is a continuous mapping, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τ_n(0) = A_n/B_n. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of Τ_n(∞) = A_n-1/B_n-1. So if the continued fraction converges the transformation Τ_n(z) maps both very small z and very large z into an arbitrarily small neighborhood of x, the value of the continued fraction, as n gets larger and larger.

What about intermediate values of z? Well, since the successive convergents are getting closer together we must have

{\frac {A_{n-1}}{B_{n-1}}}\approx {\frac {A_{n}}{B_{n}}}\quad \Rightarrow \quad {\frac {A_{n-1}}{A_{n}}}\approx {\frac {B_{n-1}}{B_{n}}}=k\,

where k is a constant, introduced for convenience. But then, by substituting in the expression for Τ_n(z) we obtain

{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}}={\frac {A_{n}}{B_{n}}}\left({\frac {z{\frac {A_{n-1}}{A_{n}}}+1}{z{\frac {B_{n-1}}{B_{n}}}+1}}\right)\approx {\frac {A_{n}}{B_{n}}}\left({\frac {zk+1}{zk+1}}\right)={\frac {A_{n}}{B_{n}}}\,

so that even the intermediate values of z (except when z ≈ −k⁻¹) are mapped into an arbitrarily small neighborhood of x, the value of the continued fraction, as n gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.^[17]

Notice that the sequence {Τ_n} lies within the automorphism group of the extended complex plane, since each Τ_n is a linear fractional transformation for which ab ≠ cd. And every member of that automorphism group maps the extended complex plane into itself – not one of the Τ_ns can possibly map the plane into a single point. Yet in the limit the sequence {Τ_n} defines an infinite continued fraction which (if i t converges) represents a single point in the complex plane.

How is this possible? Think of it this way. When an infinite continued fraction converges, the corresponding sequence {Τ_n} of LFTs "focuses" the plane in the direction of x, the value of the continued fraction. At each stage of the process a larger and larger region of the plane is mapped into a neighborhood of x, and the smaller and smaller region of the plane that's left over is stretched out ever more thinly to cover everything outside that neighborhood.^[18]

What about divergent continued fractions? Can those also be interpreted geometrically? In a word, yes. We distinguish three cases.

The two sequences {Τ_2n-1} and {Τ_2n} might themselves define two convergent continued fractions that have two different values, x_odd and x_even. In this case the continued fraction defined by the sequence {Τ_n} diverges by oscillation between two distinct limit points. And in fact this idea can be generalized – sequences {Τ_n} can be constructed that oscillate among three, or four, or indeed any number of limit points. Interesting instances of this case arise when the sequence {Τ_n} constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane.
The sequence {Τ_n} may produce an infinite number of zero denominators B_i while also producing a subsequence of finite convergents. These finite convergents may not repeat themselves or fall into a recognizable oscillating pattern. Or they may converge to a finite limit, or even oscillate among multiple finite limits. No matter how the finite convergents behave, the continued fraction defined by the sequence {Τ_n} diverges by oscillation with the point at infinity in this case.^[19]
The sequence {Τ_n} may produce no more than a finite number of zero denominators B_i. while the subsequence of finite convergents dances wildly around the plane in a pattern that never repeats itself and never approaches any finite limit, either.

Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction

x=1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{1+\ddots }}}}}}}}\,

where z is any real number such that z < −¼.^[20]

Euler's continued fraction formula

Euler proved the following identity:^[8]

a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}={\frac {a_{0}}{1-}}{\frac {a_{1}}{1+a_{1}-}}{\frac {a_{2}}{1+a_{2}-}}\cdots {\frac {a_{n}}{1+a_{n}}}.\,

From this many other results can be derived, such as

{\frac {1}{u_{1}}}+{\frac {1}{u_{2}}}+{\frac {1}{u_{3}}}+\cdots +{\frac {1}{u_{n}}}={\frac {1}{u_{1}-}}{\frac {u_{1}^{2}}{u_{1}+u_{2}-}}{\frac {u_{2}^{2}}{u_{2}+u_{3}-}}\cdots {\frac {u_{n-1}^{2}}{u_{n-1}+u_{n}}},\,

and

{\frac {1}{a_{0}}}+{\frac {x}{a_{0}a_{1}}}+{\frac {x^{2}}{a_{0}a_{1}a_{2}}}+\cdots +{\frac {x^{n}}{a_{0}a_{1}a_{2}\ldots a_{n}}}={\frac {1}{a_{0}-}}{\frac {a_{0}x}{a_{1}+x-}}{\frac {a_{1}x}{a_{2}+x-}}\cdots {\frac {a_{n-1}x}{a_{n}+x}}.\,

Euler's formula connecting continued fractions and series is the motivation for the fundamental inequalities, and also the basis of elementary approaches to the convergence problem.

Examples

Transcendental functions and numbers

Here are two continued fractions that can be built via Euler's identity.

$e^{x}={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots =1+{\cfrac {x}{1-{\cfrac {1x}{2+x-{\cfrac {2x}{3+x-{\cfrac {3x}{4+x-\ddots }}}}}}}}$

$\log(1+x)={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots ={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+\ddots }}}}}}}}$

Here are additional generalized continued fractions:

$\tan ^{-1}{\cfrac {x}{y}}={\cfrac {xy}{1y^{2}+{\cfrac {(1xy)^{2}}{3y^{2}-1x^{2}+{\cfrac {(3xy)^{2}}{5y^{2}-3x^{2}+{\cfrac {(5xy)^{2}}{7y^{2}-5x^{2}+\ddots }}}}}}}}={\cfrac {x}{1y+{\cfrac {(1x)^{2}}{3y+{\cfrac {(2x)^{2}}{5y+{\cfrac {(3x)^{2}}{7y+\ddots }}}}}}}}$

$e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}};\;e^{2}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots }}}}}}}}$

$\log \left(1+{\frac {x}{y}}\right)={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}$

This last is based on an algorithm derived by Alekseĭ Nikolaevich Khovanskiĭ in the 1970s.^[21]

Example: the natural logarithm of 2 (= [0;1,2,3,1,5,2/3,7,1/2,9,2/5,...,2k-1,2/k,...] ≈ 0.693147...):^[22]

$\log 2=\log(1+1)={\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {2}{5+{\cfrac {3}{2+\ddots }}}}}}}}}}}}={\cfrac {2}{3-{\cfrac {1^{2}}{9-{\cfrac {2^{2}}{15-{\cfrac {3^{2}}{21-\ddots }}}}}}}}$

$π$

Here are three of $π$ 's best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting x=y=1 and multiplying by four. The Leibniz formula for $π$ :

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}=\sum _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+-\cdots$

converges too slowly, requiring roughly 3 x 10ⁿ terms to achieve n-decimal precision. The series derived by Nilakantha Somayaji:

$\pi =3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+\ddots }}}}}}=3-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(n+1)(2n+1)}}=3+{\frac {1}{1\cdot 2\cdot 3}}-{\frac {1}{2\cdot 3\cdot 5}}+{\frac {1}{3\cdot 4\cdot 7}}-+\cdots$

is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both converge sublinearly to $π$ . On the other hand:

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}=4-1+{\frac {1}{6}}-{\frac {1}{34}}+{\frac {16}{3145}}-{\frac {4}{4551}}+{\frac {1}{6601}}-{\frac {1}{38341}}+-\cdots$

converges linearly to $π$ , adding at least three decimals digits of precision per four terms, a pace slightly faster than the arcsine formula for $π$ :

$\pi =6\sin ^{-1}\left({\frac {1}{2}}\right)=\sum _{n=0}^{\infty }{\frac {3\cdot {\binom {2n}{n}}}{16^{n}(2n+1)}}={\frac {3}{16^{0}\cdot 1}}+{\frac {6}{16^{1}\cdot 3}}+{\frac {18}{16^{2}\cdot 5}}+{\frac {60}{16^{3}\cdot 7}}+\cdots \!$

which adds at least three decimal digits per five terms. ^[23]

Note: Using the continued fraction for $\tan ^{-1}{\frac {x}{y}}$ cited above with the best-known Machin-like formula provides an even more rapidly converging expression:

$\pi =16\tan ^{-1}{\cfrac {1}{5}}\,-\,4\tan ^{-1}{\cfrac {1}{239}}={\cfrac {16}{u+{\cfrac {1^{2}}{3u+{\cfrac {2^{2}}{5u+{\cfrac {3^{2}}{7u+\ddots }}}}}}}}\,-\,{\cfrac {4}{v+{\cfrac {1^{2}}{3v+{\cfrac {2^{2}}{5v+{\cfrac {3^{2}}{7v+\ddots }}}}}}}}.$

where $u=5,\;v=239.$

Roots of positive numbers

The nth root of any positive number z^m can be expressed by restating z = xⁿ + y, resulting in

${\sqrt[{n}]{z^{m}}}={\sqrt[{n}]{(x^{n}+y)^{m}}}=x^{m}+{\cfrac {my}{nx^{n-m}+{\cfrac {(n-m)y}{2x^{m}+{\cfrac {(n+m)y}{3nx^{n-m}+{\cfrac {(2n-m)y}{2x^{m}+{\cfrac {(2n+m)y}{5nx^{n-m}+{\cfrac {(3n-m)y}{2x^{m}+\ddots }}}}}}}}}}}}$

which can be simplified, by folding each pair of fractions into one fraction, to

${\sqrt[{n}]{z^{m}}}=x^{m}+{\cfrac {2x^{m}\cdot my}{n(2z-y)-my-{\cfrac {(1^{2}n^{2}-m^{2})y^{2}}{3n(2z-y)-{\cfrac {(2^{2}n^{2}-m^{2})y^{2}}{5n(2z-y)-{\cfrac {(3^{2}n^{2}-m^{2})y^{2}}{7n(2z-y)-{\cfrac {(4^{2}n^{2}-m^{2})y^{2}}{9n(2z-y)-\ddots }}}}}}}}}}.$

The square root of z is a special case of this nth root algorithm (m=1, n=2):

${\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {3y}{6x+{\cfrac {3y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {1\cdot 3y^{2}}{6(2z-y)-{\cfrac {3\cdot 5y^{2}}{10(2z-y)-\ddots }}}}}}$

which can be simplified by noting that 5/10 = 3/6 = 1/2:

${\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {y^{2}}{2(2z-y)-{\cfrac {y^{2}}{2(2z-y)-\ddots }}}}}}.$

The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper x and y.

Example 1

The cube root of two (2^1/3 or ³√2 ≈ 1.259921...):

(A) "Standard notation" of x = 1, y = 1, and 2z - y = 3:

${\sqrt[{3}]{2}}=1+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {4}{9+{\cfrac {5}{2+{\cfrac {7}{15+{\cfrac {8}{2+{\cfrac {10}{21+{\cfrac {11}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{9-1-{\cfrac {2\cdot 4}{27-{\cfrac {5\cdot 7}{45-{\cfrac {8\cdot 10}{63-{\cfrac {11\cdot 13}{81-\ddots }}}}}}}}}}.$

(B) Rapid convergence with x = 5, y = 3 and 2z - y = 253:

${\sqrt[{3}]{2}}={\cfrac {5}{4}}+{\cfrac {0.5}{50+{\cfrac {2}{5+{\cfrac {4}{150+{\cfrac {5}{5+{\cfrac {7}{250+{\cfrac {8}{5+{\cfrac {10}{350+{\cfrac {11}{5+\ddots }}}}}}}}}}}}}}}}={\cfrac {5}{4}}+{\cfrac {2.5\cdot 1}{253-1-{\cfrac {2\cdot 4}{759-{\cfrac {5\cdot 7}{1265-{\cfrac {8\cdot 10}{1771-\ddots }}}}}}}}.$

Example 2

Pogson's ratio (100^1/5 or ⁵√100 ≈ 2.511886...), with x = 5, y = 75 and 2z - y = 6325:

${\sqrt[{5}]{100}}={\cfrac {5}{2}}+{\cfrac {3}{250+{\cfrac {12}{5+{\cfrac {18}{750+{\cfrac {27}{5+{\cfrac {33}{1250+{\cfrac {42}{5+\ddots }}}}}}}}}}}}={\cfrac {5}{2}}+{\cfrac {5\cdot 3}{1265-3-{\cfrac {12\cdot 18}{3795-{\cfrac {27\cdot 33}{6325-{\cfrac {42\cdot 48}{8855-\ddots }}}}}}}}.$

Example 3

The twelfth root of two (2^1/12 or ¹²√2 ≈ 1.059463...), using "standard notation":

${\sqrt[{12}]{2}}=1+{\cfrac {1}{12+{\cfrac {11}{2+{\cfrac {13}{36+{\cfrac {23}{2+{\cfrac {25}{60+{\cfrac {35}{2+{\cfrac {37}{84+{\cfrac {47}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{36-1-{\cfrac {11\cdot 13}{108-{\cfrac {23\cdot 25}{180-{\cfrac {35\cdot 37}{252-{\cfrac {47\cdot 49}{324-\ddots }}}}}}}}}}.$

Example 4

Equal temperament's perfect fifth (2^7/12 or ¹²√2⁷ ≈ 1.498307...), with m=7:

(A) "Standard notation":

${\sqrt[{12}]{2^{7}}}=1+{\cfrac {7}{12+{\cfrac {5}{2+{\cfrac {19}{36+{\cfrac {17}{2+{\cfrac {31}{60+{\cfrac {29}{2+{\cfrac {43}{84+{\cfrac {41}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 7}{36-7-{\cfrac {5\cdot 19}{108-{\cfrac {17\cdot 31}{180-{\cfrac {29\cdot 43}{252-{\cfrac {41\cdot 55}{324-\ddots }}}}}}}}}}.$

(B) Rapid convergence with x = 3, y = –7153, and 2z - y = 2¹⁹+3¹²:

${\sqrt[{12}]{2^{7}}}={\cfrac {1}{2}}{\sqrt[{12}]{3^{12}-7153}}={\cfrac {3}{2}}-{\cfrac {0.5\cdot 7153}{4\cdot 3^{12}-{\cfrac {11\cdot 7153}{6-{\cfrac {13\cdot 7153}{12\cdot 3^{12}-{\cfrac {23\cdot 7153}{6-{\cfrac {25\cdot 7153}{20\cdot 3^{12}-{\cfrac {35\cdot 7153}{6-{\cfrac {37\cdot 7153}{28\cdot 3^{12}-{\cfrac {47\cdot 7153}{6-\ddots }}}}}}}}}}}}}}}}$

${\sqrt[{12}]{2^{7}}}={\cfrac {3}{2}}-{\cfrac {3\cdot 7153}{12(2^{19}+3^{12})+7153-{\cfrac {11\cdot 13\cdot 7153^{2}}{36(2^{19}+3^{12})-{\cfrac {23\cdot 25\cdot 7153^{2}}{60(2^{19}+3^{12})-{\cfrac {35\cdot 37\cdot 7153^{2}}{84(2^{19}+3^{12})-\ddots }}}}}}}}.$

More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions.

Higher dimensions

Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be. Another reason is to find a possible solution to Hermite's problem.

There have been numerous attempts to construct a generalized theory. Notable efforts in this direction were made by Felix Klein (the Klein polyhedron), Georges Poitou and George Szekeres.

Notes

↑ Jones & Thron (1980) p.20
↑ Thomas W. Cusick; Mary E. Flahive (1989). The Markoff and Lagrange Spectra. American Mathematical Society. p. 89. ISBN 0-8218-1531-8.
↑ George Chrystal (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.
↑ 300 BC Euclid, Elements - The Euclidean algorithm generates a continued fraction as a by-product.
↑ 1579 Rafael Bombelli, L'Algebra Opera
↑ 1613 Pietro Cataldi, Trattato del modo brevissimo di trovar la radice quadra delli numeri; roughly translated, A treatise on a quick way to find square roots of numbers.
↑ 1695 John Wallis, Opera Mathematica, Latin for Mathematical Works.
1 2 1748 Leonhard Euler, Introductio in analysin infinitorum, Vol. I, Chapter 18.
↑ The Irrationals: A Story of the Numbers You Can't Count On, Julian Havil, Princeton University Press, 2012, pp.104-105
↑ Brahmagupta (598 - 670) was the first mathematician to make a systematic study of Pell's equation.
↑ 1813 Carl Friedrich Gauss, Werke, Vol. 3, pp. 134-138.
↑ 1895 Helge von Koch, Bull. Soc. Math. de France, "Sur un théorème de Stieltjes et sur les fractions continues".
↑ When z is taken to be an integer this function is quite famous; it generates the golden ratio and the closely related sequence of silver means.
↑ 1929 Oskar Perron, Die Lehre von den Kettenbrüchen derives even more general extension and contraction formulas for continued fractions.
↑ Angell, David (2007). "Irrationality and Transcendence - Lambert's Irrationality Proofs". School of Mathematics, University of New South Wales.
↑ Porubský, Štefan. "Basic definitions for continued fractions". Interactive Information Portal for Algorithmic Mathematics. Prague, Czech Republic,: Institute of Computer Science of the Czech Academy of Sciences. Retrieved 9 April 2013.
↑ This intuitive interpretation is not rigorous because a continued fraction is not a mapping – it is the limit of a sequence of mappings. This construction of an infinite continued fraction is roughly analogous to the construction of an irrational number as the limit of a Cauchy sequence of rational numbers.
↑ Because of analogies like this one, the theory of conformal mapping is sometimes described as "rubber sheet geometry".
↑ One approach to the convergence problem is to construct positive definite continued fractions, for which the denominators B_i are never zero.
↑ This periodic fraction of period one is discussed more fully in the article convergence problem.
↑ An alternative way to calculate log(x)
↑ On the Ramanujan AGM Fraction, I: The Real-Parameter Case. Experimental Mathematics, Vol. 13 (2004), No. 3, pages 278,280.
↑ Beckmann, Petr (1971). A History of Pi. St. Martin's Press, Inc. pp. 131–133, 140–143. ISBN 0-88029-418-3. . Note: this continued fraction's rate of convergence μ tends to 3 – √8 ≈ 0.1715729, hence 1/μ tends to 3 + √8 ≈ 5.828427, whose common logarithm is 0.7655... ≈ 13/17 > 3/4. The same 1/μ = 3 + √8 (the silver ratio squared) also is observed in the unfolded general continued fractions of both the natural logarithm of 2 and the nth root of 2 (which works for any integer n > 1) if calculated using 2 = 1 + 1. For the folded general continued fractions of both expressions, the rate convergence μ = (3–√8)² = 17–√288 ≈ 0.02943725, hence 1/μ = (3+√8)² = 17+√288 ≈ 33.97056, whose common logarithm is 1.531... ≈ 26/17 > 3/2, thus adding at least three digits per two terms. This is because the folded GCF folds each pair of fractions from the unfolded GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions.

References

Jones, William B.; Thron, W.J. (1980), Continued fractions. Analytic theory and applications, Encyclopedia of Mathematics and its Applications, 11, Reading, MA: Addison-Wesley, ISBN 0-201-13510-8, Zbl 0445.30003 (Covers both analytic theory and history).
Lisa Lorentzen and Haakon Waadeland, Continued Fractions with Applications, North Holland, 1992. ISBN 978-0-444-89265-2. (Covers primarily analytic theory and some arithmetic theory).
Oskar Perron, Die Lehre von den Kettenbrüchen Band I, II, B.G. Teubner, 1954.
George Szekeres, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13, "Multidimensional Continued Fractions", pp. 113–140, 1970.
H.S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973. ISBN 0-8284-0207-8. (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.)
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.2. Evaluation of Continued Fractions", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
Manny Sardina, General Method for Extracting Roots using (Folded) Continued Fractions, Surrey (UK), 2007.

External links

The first twenty pages of Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, ISBN 0-521-81805-2, contains generalized continued fractions for √2 and the golden mean.
"Sloane's A133593 : Exact continued fraction for Pi". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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