Continued fraction
A continued fraction is a mathematical expression written as a fraction whose denominator contains a sum involving another fraction, which may itself be a simple or a continued fraction. If this iteration terminates with a simple fraction, the result is a finite continued fraction; if it continues indefinitely, the result is an infinite continued fraction. The special case in which all numerators are equal to one, and all denominators are positive integers, is referred to as a simple continued fraction. Any rational number can be expressed as a finite simple continued fraction, and any irrational number can be expressed as an infinite simple continued fraction.
Different areas of mathematics use different terminology and notation for continued fractions. In number theory, the unqualified term continued fraction usually refers to simple continued fractions, whereas the general case is referred to as generalized continued fractions. In complex analysis and numerical analysis, the general case is usually referred to by the unqualified term continued fraction.
The numerators and denominators of continued fractions can be sequences of constants or functions.
Formulation
A continued fraction is an expression of the formwhere the are the partial numerators, the are the partial denominators, and the leading term is called the integer part of the continued fraction.
The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:
where is the numerator and is the denominator, called continuants, of the th convergent. They are given by the three-term recurrence relation
with initial values
If the sequence of convergents 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, or it may produce an infinite number of zero denominators.
History
The story of continued fractions begins with the Euclidean algorithm, a procedure for finding the greatest common divisor of two natural numbers and. 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 devised a technique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century. Now the pace of development quickened. Just 24 years later, in 1613, Pietro Cataldi introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as
with the dots indicating where the next fraction goes, and each representing a modern plus sign.
Late in the seventeenth century John Wallis introduced the term "continued fraction" into mathematical literature. New techniques for mathematical analysis 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. 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 is irrational, by using the following continued fraction for :
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. 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, it contains a palindromic string of length.
In 1813 Gauss derived from complex-valued hypergeometric functions what are now called Gauss's continued fractions. They can be used to express many elementary functions and some more advanced 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 easy for an unfamiliar reader to interpret. However, it takes up a lot of space and can be difficult to typeset. So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction sets each nested fraction on the same line, indicating the nesting by dangling plus signs in the denominators:Sometimes the plus signs are typeset to vertically align with the denominators but not under the fraction bars:
Pringsheim wrote a generalized continued fraction this way:
Carl Friedrich Gauss evoked the more familiar infinite product when he devised this notation:
Here the "" 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 is zero, the infinite continued fractionis really just a finite continued fraction with fractional terms, and therefore a rational function of to and to. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all. There is no need to place this restriction on the partial denominators.
The determinant formula
When the th convergent of a continued fractionis expressed as a simple fraction we can use the determinant formula
to relate the numerators and denominators of successive convergents and to one another.
The proof for this can be easily seen by induction.
Base case
'''Inductive step'''
The equivalence transformation
If is any infinite sequence of non-zero complex numbers we can prove, by induction, thatwhere 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 are zero, a sequence can be chosen to make each partial numerator a 1:
where,,, and in general.
Second, if none of the partial denominators are zero we can use a similar procedure to choose another sequence to make each partial denominator a 1:
where and otherwise.
These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.
Notions of convergence
As mentioned in the introduction, the continued fractionconverges if the sequence of convergents tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the part of the fraction by, instead of by 0, to compute the convergents. The convergents thus obtained are called modified convergents. We say that the continued fraction converges generally if there exists a sequence such that the sequence of modified convergents converges for all sufficiently distinct from. The sequence is then called an exceptional sequence for the continued fraction. See Chapter 2 of for a rigorous definition.
There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely convergent when the series
where are the convergents of the continued fraction, converges absolutely. The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence.
Finally, a continued fraction of one or more complex variables is uniformly convergent in an open neighborhood when its convergents converge uniformly on ; that is, when for every there exists such that for all, for all,
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 and, then the sequence must converge to one of these, and 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, and the other converging to.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
is a continued fraction, then the even part and the odd part are given by
and
respectively. More precisely, if the successive convergents of the continued fraction are, then the successive convergents of as written above are, and the successive convergents of are.
Conditions for irrationality
If and are positive integers with for all sufficiently large, thenconverges to an irrational limit.
Fundamental recurrence formulas
The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas:The continued fraction's successive convergents are then given by
These recurrence relations are due to John Wallis and Leonhard Euler.
These recurrence relations are simply a different notation for the relations obtained by Pietro Antonio Cataldi.
As an example, consider the simple continued fraction in canonical form that represents the golden ratio :
Applying the fundamental recurrence formulas we find that the successive numerators are and the successive denominators are, 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.