1911 Encyclopædia Britannica/Continued Fractions
CONTINUED FRACTIONS. In mathematics, an expression of the form
a_{1} ± | b_{2} | |||
a_{2} ± | b_{3} | |||
a_{3} ± | b_{4} | |||
a_{4} ± | b_{5} | |||
a_{5} ± | . . ., |
where a_{1}, a_{2}, a_{3}, ... and b_{2}, b_{3}, b_{4}, ... are any quantities whatever, positive or negative, is called a “continued fraction.” The quantities a_{1} ..., b_{2} ... may follow any law whatsoever. If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a_{1} ..., b_{2} ... is infinite it is said to be a non-terminating or infinite continued fraction. If b_{2}/a_{2}, b_{3}/a_{3} ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number.
The notation employed by English writers for the general continued fraction is
a_{1} ± | b_{2} | b_{3} | b_{4} | ... | |||
a_{2} | ± | a_{2} | ± | a_{2} | ± |
Continental writers frequently use the notation
a_{1} ± | b_{2} | ± | b_{3} | ± | b_{4} | ± ..., or a_{1} ± | b_{2} | ± | b_{3} | ± | b_{4} | ± ... | |||
a_{2} | a_{3} | a_{4} | a_{2} | a_{3} | a_{4} |
The terminating continued fractions
a_{1}, a_{1} + | b_{2} | , a_{1} + | b_{2} | b_{3} | , a_{1} + | b_{2} | b_{3} | b_{4} | , ... | ||||
a_{2} | a_{2} | + | a_{3} | a_{2} | + | a_{3} | + | a_{4} |
reduced to the forms
a_{1} | , | a_{1}a_{2} + b_{2} | , | a_{1}a_{2}a_{3} + b_{2}a_{3} + b_{2}a_{1} | , | a_{1}a_{2}a_{3}a_{4} + b_{2}a_{3}a_{4} + b_{3}a_{1}a_{4} + b_{4}a_{1}a_{2} + b_{2}b_{4} | , ... |
1 | a_{2} | a_{2}a_{3} + b_{3} | a_{2}a_{3}a_{4} + a_{4}b_{3} + a_{2}b_{4} |
are called the successive convergents to the general continued fraction.
Their numerators are denoted by p_{1}, p_{2}, p_{3}, p_{4}...; their denominators by q_{1}, q_{2}, q_{3}, q_{4}....
We have the relations
In the case of the fraction
a_{1} − | b_{2} | b_{3} | b_{4} | ..., | |||
a_{2} | − | a_{3} | − | a_{4} | − |
we have the relations p_{n} = a_{n}p_{n−1} − b_{n}p_{n−2}, q_{n}= a_{n}q_{n−1} − b_{n}q_{n−2}.
Taking the quantities a_{1} ..., b_{2} ... to be all positive, a continued fraction of the form
a_{1} + | b_{2} | b_{3} | ..., | ||
a_{2} | + | a_{3} | + |
is called a continued fraction of the first class; a continued fraction of the form
b_{2} | b_{3} | b_{4} | ... | |||
a_{2} | − | a_{3} | − | a_{4} | − |
is called a continued fraction of the second class.
A continued fraction of the form
a_{1} + | 1 | 1 | 1 | ..., | |||
a_{2} | + | a_{3} | + | a_{4} | + |
where a_{1}, a_{2}, a_{3}, a_{4} ... are all positive integers, is called a simple continued fraction. In the case of this fraction a_{1}, a_{2}, a_{3}, a_{4} ... are called the successive partial quotients. It is evident that, in this case,
are two series of positive integers increasing without limit if the fraction does not terminate.
The general continued fraction
a_{1} + | b_{2} | b_{3} | b_{4} | ... | |||
a_{2} | + | a_{3} | + | a_{4} | + |
is evidently equal, convergent by convergent, to the continued fraction
a_{1} + | λ_{2}b_{2} | λ_{2}λ_{3}b_{3} | λ_{3}λ_{4}b_{4} | ..., | |||
λ_{2}a_{2} | + | λ_{3}a_{3} | + | λ_{4}a_{4} | + |
where λ_{2}, λ_{3}, λ_{4}, ... are any quantities whatever, so that by choosing λ_{2}b_{2} = 1, λ_{2}λ_{3}b_{3} = 1, &c., it can be reduced to any equivalent continued fraction of the form
a_{1} + | 1 | 1 | 1 | ..., | |||
d_{2} | + | d_{3} | + | d_{4} | + |
Simple Continued Fractions.
1. The simple continued fraction is both the most interesting and important kind of continued fraction.
Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable.
In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, a_{n} as a_{n} − 1 + 11.
The numerators and denominators of the successive convergents obey the law p_{n}q_{n−1} − p_{n−1}q_{n} = (−1)^{n}, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:—
The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.
Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.
Every convergent is nearer to the value of the whole fraction than any preceding convergent.
Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.
The difference between the continued fraction and the n^{th} convergent is less than 1q_{n}q_{n+1} and greater than a_{n+2}q_{n}q_{n+2}. These limits may be replaced by the following, which, though not so close, are simpler, viz. 1q^{2}_{n} and 1q_{n}(q_{n} + q_{n+1})
Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since
p_{n} | ~ | p_{n−1} | = | 1 | , Lt. | p_{n} | = Lt. | p_{n−1} | , |
q_{n} | q_{n−1} | q_{n}q_{n−1} | q_{n} | q_{n−1} |
so that its value cannot oscillate.
The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our approximation. Thus a continued fraction equivalent to π (the ratio of the circumference to the diameter of a circle) is
3 + | 1 | 1 | 1 | 1 | 1 | 1 | ... | ||||||
7 | + | 15 | + | 1 | + | 292 | + | 1 | + | 1 | + |
of which the successive convergents are
3 | , | 22 | , | 333 | , | 355 | , | 103993 | , &c., |
1 | 7 | 106 | 113 | 33102 |
the fourth of which is accurate to the sixth decimal place, since the error lies between 1/q_{4}q_{5} or .0000002673 and a_{6}/q_{4}q_{6} or .0000002665.
Similarly the continued fraction given by Euler as equivalent to 12(e − 1) (e being the base of Napierian logarithms), viz.
1 | 1 | 1 | 1 | 1 | |||||
1 | + | 6 | + | 10 | + | 14 | + | 18 | + ..., |
may be used to approximate very rapidly to the value of e.
For the application of continued fractions to the problem “To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable,” the reader is referred to G. Chrystal’s Algebra, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar (q.v.).
Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. Such a method is, however, too tedious, compared with such a method as Homer’s, to be of any practical value.
The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. If we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq − bp = +1 or −1, according as the number of convergents is even or odd, which we can take them to be as we please. If we take aq−bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq − bt, y = at − cp; if we take aq − bp = −1, we have x = bt − cq, y = cp − at.
An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. 2 of the Acta Mathematica.
Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. A. Serret’s Cours d’Algèbre Supérieure.
2. Recurring Simple Continued Fractions.—The infinite continued fraction
a_{1} + | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||
a_{2} | + | a_{3} | + ... + | a_{n} | + | b_{1} | + | b_{2} | + ... + | b_{n} | + | b_{1} | + | b_{2} | + ... + | b_{n} | + | b_{1} | + ..., |
where, after the n^{th} partial quotient, the cycle of partial quotients b_{1}, b_{2}, ..., b_{n} recur in the same order, is the type of a recurring simple continued fraction.
The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Thus
2 − √3 = | 1 | 1 | 1 | 1 | 1 | |||||
3 | + | 1 | + | 2 | + | 1 | + | 2 | + ..., |
√28 = 5 + | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
3 | + | 2 | + | 3 | + | 10 | + | 3 | + | 2 | + | 3 | + | 10 | + ... |
The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. There is only one non-recurring partial quotient a_{1}. If b_{1}, b_{2}, ..., b_{n} is the cycle of recurring quotients, then b_{n} = 2a_{1}, b_{1} = b_{n−1}, b_{2} = b_{n−2}, b_{3} = b_{n-3}, &c.
In the case of a recurring continued fraction which represents √N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^{2}_{nr} −Nq^{2}_{nr} = (−1)^{nr}, whence, if n is odd, integral solutions of the indeterminate equation x^{2} − Ny^{2} = ±1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x^{2} −Ny^{2} = +1 can be found.
The theory and development of the simple recurring continued fraction is due to Lagrange. For proofs of the theorems here stated and for applications to the more general indeterminate equation x^{2} −Ny^{2} = H the reader may consult Chrystal’s Algebra or Serret’s Cours d’Algèbre Supérieure; he may also profitably consult a tract by T. Muir, The Expression of a Quadratic Surd as a Continued Fraction (Glasgow, 1874).
1. The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation u_{n} = a_{n}u_{n−1} + b_{n}u_{n−2}. When we can solve this equation we have an expression for the n^{th} convergent to the fraction, generally in the form of the quotient of two series, each of n terms. As an example, take the fraction (known as Brouncker’s fraction, after Lord Brouncker)
1 | 1^{2} | 3^{2} | 5^{2} | 7^{2} | |||||
1 | + | 2 | + | 2 | + | 2 | + | 2 | + ... |
Here we have
whence
and we readily find that
p_{n} | = 1 − | 1 | + | 1 | – | 1 | + ... ± | 1 | , |
q_{n} | 3 | 5 | 7 | 2n + 1 |
whence the value of the fraction taken to infinity is 14π.
It is always possible to find the value of the n^{th} convergent to a recurring continued fraction. If r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p’s and q’s obtain a linear relation connecting
in which the coefficients are all constants. Or we may proceed as follows. (We need not consider a fraction with a non-recurring part). Let the fraction be
a_{1} | a_{2} | a_{r} | a_{1} | ||||
b_{1} | + | b_{2} | + ... + | b_{r} | + | b_{1} | + ... |
Let u_{n} ≡ | p_{nr+m } | ; then u_{n} = | a_{1} | a_{2} | a_{r} | , | ||
q_{nr+m} | b_{1} | + | b_{2} | + ... + | b_{r} + u_{n1} |
leading to an equation of the form Au_{n}u_{n−1} + Bu_{n} + Cu_{n−1} + D = 0, where A, B, C, D are independent of n, which is readily solved.
2. The Convergence of Infinite Continued Fractions.—We have seen that the simple infinite continued fraction converges. The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. It may, however, oscillate. We have the relation p_{n}q_{n−1} − p_{n−1}q_{n} = (−1)^{n}b_{2}b_{3}...b_{n}, from which
p_{n} | – | p_{n−1} | = (−1)^{n} | b_{2}b_{3} ... b_{n} | , |
q_{n} | q_{n−1} | q_{n}q_{n−1} |
and the limit of the right-hand side is not necessarily zero.
The tests for convergency are as follows:
Let the continued fraction of the first class be reduced to the form
d_{1} + | 1 | 1 | 1 | |||
d_{2} | + | d_{3} | + | d_{4} | + ..., |
then it is convergent if at least one of the series d_{3} + d_{5} + d_{7} + ..., d_{2} + d_{4} + d_{6} + ... diverges, and oscillates if both these series converge.
For the convergence of the continued fraction of the second class there is no complete criterion. The following theorem covers a large number of important cases.
“If in the infinite continued fraction of the second class a_{n} ≥ b_{n} + 1 for all values of n, it converges to a finite limit not greater than unity.”
3. The Incommensurability of Infinite Continued Fractions.—There is no general test for the incommensurability of the general infinite continued fraction.
Two cases have been given by Legendre as follows:—
If a_{2}, a_{3}, ..., a_{n}, b_{2}, b_{3}, ..., b_{n} are all positive integers, then
I. The infinite continued fraction
b_{2} | b_{3} | b_{n} | |||
a_{2} | + | a_{3} | + ... + | a_{n} | + ... |
converges to an incommensurable limit if after some finite value of n the condition a_{n} ≠ b_{n} is always satisfied.
II. The infinite continued fraction
b_{2} | b_{3} | b_{n} | |||
a_{2} | − | a_{3} | − ... − | a_{n} | − ... |
converges to an incommensurable limit if after some finite value of n the condition a_{n} ≥ b_{n} + 1 is always satisfied, where the sign > need not always occur but must occur infinitely often.
The functions p_{n} and q_{n}, regarded as functions of a_{1}, ..., a_{n}, b_{2}, ..., b_{n} determined by the relations
q_{n} = a_{n}q_{n−1} + b_{n}q_{n−2},
with the conditions p_{1} = a_{1}, p_{0} = 1; q_{2} = a_{2}, q_{1} = 1, q_{0} = 0, have been studied under the name of continuants. The notation adopted is
p_{n} = K | ( | a_{1}, | b_{2}, ..., b_{n} a_{2}, ..., a_{n} | ), |
and it is evident that we have
q_{n} = K | ( | a_{1}, | b_{3}, ..., b_{n} a_{3}, ..., a_{n} | ). |
The theory of continuants is due in the first place to Euler. The reader will find the theory completely treated in Chrystal’s Algebra, where will be found the exhibition of a prime number of the form 4p + 1 as the actual sum of two squares by means of continuants, a result given by H. J. S. Smith.
The continuant K ( b_{2}, b_{3}, ..., b_{n}
a_{1}, a_{2}, a_{3}, ..., a_{n}) is also equal to the determinant
a_{1} −1 0 0
0 |
b_{2} a_{2} −1 0
0 |
0 b_{3} a_{3} −1
— |
0 0 b_{4} a_{4}
— |
. . . b_{5}
u 0 |
. . . .
−1 0 |
. . . .
a_{n−1} −1 |
0 0 0 —
b_{n} a_{n} |
from which point of view continuants have been treated by W. Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see T. Muir’s Theory of Determinants, chap. iii.).
Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange’s investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol. i. chap. iv.).
1. A continued fraction may always be found whose n^{th} convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. In fact, a continued fraction
b_{1} | b_{2} | b_{n} | |||
a_{1} | + | a_{2} | + ... + | a_{n} | + ... |
can be constructed having for the numerators of its successive convergents any assigned quantities p_{1}, p_{2}, p_{3}, ..., p_{n}, and for their denominators any assigned quantities q_{1}, q_{2}, q_{3}, ..., q_{n} ...
The partial fraction b_{n}/a_{n} corresponding to the n^{th} convergent can be found from the relations
and the first two partial quotients are given by
If we form then the continued fraction in which p_{1}, p_{2}, p_{3}, ..., p_{n} are u_{1}, u_{1} + u_{2}, u_{1} + u_{2} + u_{3}, ..., u_{1} + u_{2} + ..., u_{n}, and q_{1}, q_{2}, q_{3}, ..., q_{n} are all unity, we find the series u_{1} + u_{2} + ..., u_{n} equivalent to the continued fraction
u_{1} | u_{2} ⁄ u_{1} | u_{3} ⁄ u_{2} | u_{n} ⁄ u_{n−1} | ||||||
1 | − | 1 + | u_{2} | – | 1 + | u_{3} | – ... − | 1 + | u_{n} |
u_{1} | u_{2} | u_{n−1} |
which we can transform into
u_{1} | u_{2} | u_{1}u_{3} | u_{2}u_{4} | u_{n-2}u_{n} | , | ||||
1 | − | u_{1} + u_{2} | − | u_{2} + u_{3} | − | u_{3} + u_{4} | − ... − | u_{n−1} + u_{n} |
a result given by Euler.
2. In this case the sum to n terms of the series is equal to the n^{th} convergent of the fraction. There is, however, a different way in which a Series may be represented by a continued fraction. We may require to represent the infinite convergent power series a_{0} + a_{1}x + a_{2}x² + ... by an infinite continued fraction of the form
β_{0} | β_{1}x | β_{2}x | β_{3}x | ||||
1 | − | 1 | − | 1 | − | 1 | − ... |
Here the fraction converges to the sum to infinity of the series. Its n^{th} convergent is not equal to the sum to n terms of the series. Expressions for β_{0}, β_{1}, β_{2}, ... by means of determinants have been given by T. Muir (Edinburgh Transactions, vol. xxvii.).
A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. It is practically identical with that of finding the greatest common measure of two polynomials. As an instance leading to results of some importance consider the series
F(n,x) = 1 + | x | + | x² | + ... |
(γ + n)1! | (γ + n)(γ + n + 1)2! |
We have
F(n + 1,x) − F(n,x) = − | x | F(n + 2,x), |
(γ + n)(γ + n + 1)2! |
whence we obtain
F(1,x) | = | 1 | x ⁄ γ(γ + 1) | x ⁄ (γ + 1)(γ + 2) | |||
F(0,x) | 1 | + | 1 | + | 1 | + ..., |
which may also be written
γ | x | x | |||
γ | + | γ + 1 | + | γ + 2 | + ... |
By putting ± x² ⁄ 4 for x in F(0,x) and F(1,x), and putting at the same time γ = 1 ⁄ 2, we obtain
tan x = | x | x² | x² | x² | tanh x = | x | x² | x² | x² | ||||||||
1 | − | 3 | − | 5 | − | 7 | − ... | 1 | + | 3 | + | 5 | + | 7 | + ... |
These results were given by Lambert, and used by him to prove that π and π^{2} incommensurable, and also any commensurable power of e.
Gauss in his famous memoir on the hypergeometric series
F(α, β, γ, x) = 1 + | α · β | x + | α(α + 1)β(β + 1) | x² + ... |
1 · γ | 1 · 2 · γ · (γ + 1) |
gave the expression for F(α, β + 1, γ + 1, x) ÷ F(α, β, γ, x) as a continued fraction, from which if we put β = 0 and write γ − 1 for γ, we get the transformation
1 + | α | x + | α(α + 1) | x^{2} + | α(α + 1)(α + 2) | x^{3} + ... = | 1 | β_{1}x | β_{2}x | |||
γ | γ(γ + 1) | γ(γ + 1)(γ + 2) | 1 | − | 1 | − | 1 | − ... |
where
β_{1} = | α | , β_{3} = | (α + 1)γ | , ..., β_{2n−1} = | (α + n − 1)(γ + n − 2) | , |
γ | (γ + 1)(γ + 2) | (γ + 2n – 3)(γ + 2n − 2) |
β_{2} = | γ − α | , β_{4} = | 2(γ + 1 − α) | , ..., β_{2n} = | n(γ + n − 1 − α) | . |
γ(γ + 1) | (γ + 2)(γ + 3) | (γ + 2n – 2)(γ + 2n − 1) |
From this we may express several of the elementary series as continued fractions; thus taking α = 1, γ = 2, and putting x for -x, we have
log(1 + x) = | x | 1^{2}x | 1^{2}x | 2^{2}x | 2^{2}x | 3^{2}x | 3^{2}x | |||||||
1 | + | 2 | + | 3 | + | 4 | + | 5 | + | 6 | + | 7 | + ... |
Taking γ = 1, writing x ⁄ αalpha; for x and increasing α indefinitely, we have
e^{x = } | 1 | x | x | x | x | x | ||||||
1 | − | 1 | + | 2 | − | 3 | + | 2 | − | 5 | + ... |
For some recent developments in this direction the reader may consult a paper by L. J. Rogers in the Proceedings of the London Mathematical Society (series 2, vol. 4).
There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is
b_{4} + | b_{5} + | |||
b_{3} + | a_{5} | |||
b_{2} + | a_{4} | |||
a_{1} + | a_{3} | |||
a_{2} |
It is obviously equal to the series
a_{1} + | b_{2} | + | b_{3} | + | b_{4} | + | b_{5} | + ... |
a_{2} | a_{2}a_{3} | a_{2}a_{3}a_{4} | a_{2}a_{3}a_{4}a_{5} |
The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626. He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. His chief advance on Bombelli was in his notation. They next appear to have been used by Daniel Schwenter (1585–1636) in a Geometrica Practica published in 1618. He uses them for approximations. The theory, however, starts with the publication in 1655 by Lord Brouncker of the continued fraction
1 | 1^{2} | 3^{2} | 5^{2} | ||||
1 | + | 2 | + | 2 | + | 2 | + ... |
as an equivalent of π ⁄ 4. This he is supposed to have deduced, no one knows how, from Wallis’ formula for 4 ⁄ π viz.
3 . 3 . 5 . 5 . 7 . 7 ... |
2 . 4 . 4 . 6 . 6 . 8 ... |
John Wallis, discussing this fraction in his Arithmetica Infinitorum (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682–1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler’s Algebra (1795). Moritz A. Stern wrote at length on the subject in Crelle’s Journal (x., 1833; xi., 1834; xviii., 1838). The theory of the convergence of continued fractions is due to Oscar Schlömilch, P. F. Arndt, P. L. Seidel and Stern. O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements.
References.—For the further history of continued fractions we may refer the reader to two papers by Gunther and A. N. Favaro, Bulletins di bibliographia e di storia delle scienze mathematische e fisicke, t. vii., and to M. Cantor, Geschichte der Mathematik, 2nd Bd. For text-books treating the subject in great detail there are those of G. Chrystal in English; Serret’s Cours d`algèbre supérieure in French; and in German those of Stern, Schlömilch, Hatterdorff and Stolz. For the application of continued fractions to the theory of irrational numbers there is P. Bachmann’s Vorlesungen über die Natur der Irrationalzahnen (1892). For the application of continued fractions to the theory of lenses, see R. S. Heath’s Geometrical Optics, chaps. iv. and v. For an exhaustive summary of all that has been written on the subject the reader may consult Bd. 1 of the Encyklopädie der mathematischen Wissenschaften (Leipzig). (A. E. J.)