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 complex values. A generalized continued fraction is an expression of the form where 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 recursion 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 (for example, the odd and even convergents may approach two different limits), 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 Rafael Bombelli 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 (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. 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. 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 , it contains a palindromic string of length . In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions. 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 is 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: 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 fraction is 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 fraction is 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, that 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 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. Simple convergence concepts. It has already been noted that the continued fraction converges if the sequence of convergents 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 diverges by oscillation if the series is absolutely convergent. Sometimes the partial numerators and partial denominators of a continued fraction are expressed as functions of a complex variable . For example, a relatively simple function might be defined as 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 if the fraction's convergents converge uniformly at every point in . Or, more precisely: if, for every an integer can be found such that the  absolute value of the difference is less than for every point in an open neighborhood whenever , the continued fraction defining is uniformly convergent on . (Here denotes the th convergent of the continued fraction, evaluated at the point inside , and is the value of the infinite continued fraction at the point .) 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 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 , then converges 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 (1616–1703) and Leonhard Euler (1707–1783). As an example, consider the regular 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. Linear fractional transformations. A linear fractional transformation (LFT) is a complex function of the form where is a complex variable, and are arbitrary complex constants such that . An additional restriction that is customarily imposed, to rule out the cases in which 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. The continued fraction as a composition of LFTs. Consider a sequence of simple linear fractional transformations Here we use to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol to represent the composition of transformations ; that is, and so forth. By direct substitution from the first set of expressions into the second we see that and, in general, where the last partial denominator in the finite continued fraction is understood  to be . And, since , the image of the point under the iterated LFT is indeed the value of the finite continued fraction with partial numerators: A geometric interpretation. Defining a finite continued fraction as the image of a point under the iterated linear functional transformation leads to an intuitively appealing geometric interpretation of infinite continued fractions. The relationship can be understood by rewriting and in terms of the fundamental recurrence formulas: In the first of these equations the ratio tends toward as tends toward zero. In the second, the ratio tends toward as tends to infinity. This leads us to  our first geometric interpretation. If the continued fraction converges, the successive convergents are eventually arbitrarily close together. Since the linear fractional transformation is a continuous mapping, there must be a neighborhood of that is mapped into an arbitrarily small neighborhood of . Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of . So if the continued fraction converges the transformation maps both very small and very large into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. For intermediate values of , since the successive  convergents are getting closer together we must have where is a constant, introduced for convenience. But then, by substituting in the expression for we obtain so that even the intermediate values of (except when ) are mapped into an arbitrarily small neighborhood of , the value of the continued fraction, as gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point. Notice that the sequence lies within the automorphism group of the extended complex plane, since each is a linear fractional transformation for which . And every member of  that automorphism group maps the extended complex plane into itself: not one of the can possibly map the plane into a single point. Yet in the limit the sequence defines an infinite continued fraction which (if it converges) represents a single point in the complex plane. When an infinite continued fraction converges, the corresponding sequence of LFTs "focuses" the plane in the direction of , 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 , 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. For divergent continued fractions, we can distinguish three cases: Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction where is any real number such that . Euler's continued fraction formula. Euler proved the following identity: From this many other results can be derived, such as and Euler's formula connecting continued fractions and series is the motivation for the , 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. Here are additional generalized continued fractions: This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s. Example: the natural logarithm of 2 (= ≈ 0.693147...): Here are three of 's best-known generalized continued fractions, the first and third of which are derived from their respective formulas above by setting and multiplying by 4. The Leibniz formula for: converges too slowly, requiring roughly terms to achieve correct decimal places. The series derived by Nilakantha Somayaji: 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 . On the other hand: converges "linearly" to , adding at least three digits of precision per four terms, a pace slightly faster than the arcsine formula for: which adds at least three decimal digits per five terms. with and . Roots of positive numbers. The th root of any positive number can be expressed by restating , resulting in which can be simplified, by folding each pair of fractions into one fraction, to The square root of is a special case with and : which can be  simplified by noting that : The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper and . Example 1. The cube root of two (2 or ≈ 1.259921...) can be calculated in two ways: Firstly, "standard notation" of , , and : Secondly, a rapid convergence with , and : Example 2. Pogson's ratio (100 or ≈ 2.511886...), with , and : Example 3. The twelfth root of two (2 or ≈ 1.059463...), using "standard notation": Example 4. Equal temperament's perfect fifth (2 or ≈ 1.498307...), with : With "standard notation": A rapid convergence with , , and  : 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 , and the way lattice points in two dimensions lie to either side of the line . 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.