Continued fraction of Gauss

Continued fraction of Gauss

In complex analysis, the continued fraction of Gauss is a particular continued fraction derived from the hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions.

History

Lambert published several examples of continued fractions in this form in 1768, and both Euler and Lagrange investigated similar constructions, [Jones & Thron (1980) p. 5] but it was Carl Friedrich Gauss who utilized the clever algebraic trick described in the next section to deduce the general form of this continued fraction, in 1813. [C. F. Gauss (1813), "Werke", vol. 3 pp. 134-138.]

Although Gauss gave the form of this continued fraction, he did not give a proof of its convergence properties. Bernhard Riemann [B. Riemann (1863), "Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita" in "Werke". pp. 400-406. (Posthumous fragment).] and Thomé [L. W. Thomé (1867), "Über die Kettenbrüchentwicklung des Gaussen quotienten …," "Jour. für Math." vol. 67 pp. 299-309.] obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, by Edward Burr Van Vleck. [E. B. Van Vleck (1901), "On the convergence of the continued fraction of Gauss and other continued fractions." "Annals of Mathematics", vol. 3 pp. 1-18.]

Derivation

By substitution in the power series expansion of the classic hypergeometric series [This function, commonly written as 2"F"1("a", "b"; "c"; "z") in modern texts, is written as "F"("a", "b"; "c"; "z") in this article as a matter of notational convenience.]
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By substituting the second equation into the first, and then the first equation into the second, again and again, we obtain the continued fraction of Gauss: [Wall, 1973 (p. 337)]

:frac{F(a,b+1;c+1;z)}{F(a,b;c;z)} = cfrac{1}{1 - cfrac{frac{a(c-b)}{c(c+1)}z}{1 - cfrac{frac{(b+1)(c-a+1)}{(c+1)(c+2)}z}{1 - cfrac{frac{(a+1)(c-b+1)}{(c+2)(c+3)}z}{1 - cfrac{frac{(b+2)(c-a+2)}{(c+3)(c+4)}z}{1 - cfrac{frac{(a+2)(c-b+2)}{(c+4)(c+5)}z}{1 - ddots.

Convergence properties

Provided only that math|c is not zero or a negative integer, the continued fraction of Gauss converges almost everywhere in the complex plane. Specifically, it can be shown that
* The continued fraction of Gauss is equal to the function

::f(z) = frac{F(a,b+1;c+1;z)}{F(a,b;c;z)}

:everywhere inside the unit circle;
* The continued fraction of Gauss represents the analytic continuation of the function math|f(z) throughout the cut complex plane, where the cut extends along the positive real axis, from math|+1 to the point at infinity;
* The function represented by the continued fraction of Gauss is meromorphic; and
* The continued fraction of Gauss converges uniformly on every bounded closed region exterior to the cut (excluding those isolated points at which it has poles). [Wall, 1973 (p. 339)]

Notice also that if math|a is zero or a negative integer, or if math|b is a negative integer, the continued fraction of Gauss terminates after a finite number of partial quotients; in this case it represents a rational function of math|z. If math|a is neither zero nor a negative integer, and math|b is not a negative integer, the continued fraction of Gauss represents a transcendental function.

Extensions and modifications

Based on the hypergeometric function

The hypergeometric series "F"("a", 0; "c"; "z") is equal to unity. By setting "b" to zero and writing "c" for "c" + 1, a simplified version of the continued fraction of Gauss can be derived:

:F(a,1;c;z) = cfrac{1}{1 - cfrac{frac{a}{c}z}{1 - cfrac{frac{c-a}{c(c+1)}z}{1 - cfrac{frac{c(a+1)}{(c+1)(c+2)}z}{1 - cfrac{frac{2(c-a+1)}{(c+2)(c+3)}z}{1 - cfrac{frac{(c+1)(a+2)}{(c+3)(c+4)}z}{1 - cfrac{frac{3(c-a+2)}{(c+4)(c+5)}z}{1 - ddots}.

Based on Kummer’s confluent hypergeometric function

There are other ways to obtain modified versions of the continued fraction of Gauss. The derivation procedure can be applied to Kummer’s confluent hypergeometric function [The function math|1F1 is sometimes written as math|M(a; c; z), or (in both Wall (1973) and Jones & Thron (1980)) as math|Φ(a; c; z).]

:,_1F_1(a;c;z) = 1 + frac{a}{c,1!}z + frac{a(a+1)}{c(c+1),2!}z^2 + frac{a(a+1)(a+2)}{c(c+1)(c+2),3!}z^3 + cdots

to obtain the formula

:frac{,_1F_1(a+1;c+1;z)}{,_1F_1(a;c;z)} = cfrac{1}{1- cfrac{frac{c-a}{c(c+1)}z}{1 + cfrac{frac{a+1}{(c+1)(c+2)}z}{1 - cfrac{frac{c-a+1}{(c+2)(c+3)}z}{1 + cfrac{frac{a+2}{(c+3)(c+4)}z}{1 - cfrac{frac{c-a+2}{(c+4)(c+5)}z}{1 + - ddots.

Since math|1F1(a; c; z) is an entire function of math|z (provided that math|c ≠ 0), it can be shown that this version of the continued fraction of Gauss converges uniformly on every bounded closed region that contains no poles of the function to which it corresponds. [Jones & Thron (1980) p. 206]

Since math|1F1(0; c; z) is equal to unity, this formula can also be simplified. Substituting math|a = 0 in the preceding formula, replacing math|c + 1 by math|c, and applying an equivalence transformation produces the identity

:,_1F_1(1;c;z) = cfrac{1}{1 - cfrac{z}{c + cfrac{z}{c + 1 - cfrac{c z}{c + 2 + cfrac{2z}{c + 3 - cfrac{(c+1)z}{c + 4 + cfrac{3z}{c + 5 - cfrac{(c+2)z}{c + 6 +- ddots,

which is valid throughout the entire complex plane.

Based on the confluent hypergeometric function 0"F"1

Another confluent hypergeometric function is defined by the series [Both Wall (1973) and Jones & Thron (1980) refer to this function as Ψ("a"; "z").]

:,_0F_1(a;z) = 1 + frac{1}{a,1!}z + frac{1}{a(a+1),2!}z^2 + frac{1}{a(a+1)(a+2),3!}z^3 + cdots

By entirely analogous arguments it can be shown that the continued fraction of Gauss becomes

:frac{,_0F_1(a+1;z)}{,_0F_1(a;z)} = cfrac{1}{1 + cfrac{frac{1}{a(a+1)}z}{1 + cfrac{frac{1}{(a+1)(a+2)}z}{1 + cfrac{frac{1}{(a+2)(a+3)}z}{1 + ddots

and that this expansion converges uniformly to the meromorphic function defined by the ratio of the two convergent series (provided, of course, that "a" is neither zero nor a negative integer).

Applications

Of the classic hypergeometric function "F"

It is easily shown that the Taylor series expansion of arctan "z" in a neighborhood of zero is given by

:arctan z = zF({scriptstylefrac{1}{2,1;{scriptstylefrac{3}{2;-z^2).

The continued fraction of Gauss can be applied to this identity, yielding the expansion

:arctan z = cfrac{z}{1 + cfrac{z^2}{3 + cfrac{(2z)^2}{5 + cfrac{(3z)^2}{7 + cfrac{(4z)^2}{9 + ddots},

which converges to the principal branch of the inverse tangent function on the cut complex plane, with the cut extending along the imaginary axis from "i" to the point at infinity, and from −"i" to the point at infinity. [Wall (1973) p. 343. Notice that "i" and −"i" are branch points for the inverse tangent function.]

This particular continued fraction converges fairly quickly when "z" = 1, giving the value π/4 to seven decimal places by the ninth convergent. The corresponding series

:frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + - dots

converges much more slowly, with more than a million terms needed to yield seven decimal places of accuracy. [Jones & Thron (1980) p. 202.]

Variations of this argument can be used to produce continued fraction expansions for the natural logarithm, the arcsin function, and the generalized binomial series.

Of Kummer's confluent hypergeometric function

The error function erf ("z"), given by

:operatorname{erf}(z) = frac{2}{sqrt{piint_0^z e^{-t^2} dt,

can also be computed in terms of Kummer's hypergeometric function:

:operatorname{erf}(z) = frac{2z}{sqrt{pi e^{-z^2} ,_1F_1(1;{scriptstylefrac{3}{2;z^2).

By applying the continued fraction of Gauss, a useful expansion valid for every complex number "z" can be obtained: [Jones & Thron (1980) p. 208.]

:frac{sqrt{pi{2} e^{z^2} operatorname{erf}(z) = cfrac{z}{1 - cfrac{z^2}{frac{3}{2} +cfrac{z^2}{frac{5}{2} - cfrac{frac{3}{2}z^2}{frac{7}{2} + cfrac{2z^2}{frac{9}{2} -cfrac{frac{5}{2}z^2}{frac{11}{2} + cfrac{3z^2}{frac{13}{2} -cfrac{frac{7}{2}z^2}{frac{15}{2} + - ddots.

A similar argument can be made to derive continued fraction expansions for the Fresnel integrals, for the Dawson function, and for the incomplete gamma function. A simpler version of the argument yields two useful continued fraction expansions of the exponential function. [See the example in the article Padé table for the expansions of "ez" as continued fractions of Gauss.]

Of the confluent hypergeometric function 0"F"1

J. H. Lambert showed that

:frac{e^z - e^{-z{e^z + e^{-z = frac{,_1F_1({scriptstylefrac{3}{2;{scriptstylefrac{z^2}{4)}{,_1F_1({scriptstylefrac{1}{2;{scriptstylefrac{z^2}{4)},

from which the following continued fraction expansion of the hyperbolic tangent function is easily derived: [Wall (1973) p. 349. This particular expansion is known as Lambert's continued fraction and dates back to 1768.]

: anh z = cfrac{z}{1 + cfrac{z^2}{3 + cfrac{z^2}{5 + cfrac{z^2}{7 + ddots.

This expansion is valid for every complex number "z". Since tan "z" = −"i" tanh "iz", the continued fraction of Gauss also gives a representation of the ordinary tangent function:

: an z = cfrac{z}{1 - cfrac{z^2}{3 - cfrac{z^2}{5 - cfrac{z^2}{7 - ddots.

This formula is also valid for every complex "z".

Notes

References

*cite book|last = Jones|first = William B.|coauthors = Thron, W. J.|title = Continued Fractions: Theory and Applications|publisher = Addison-Wesley Publishing Company|location = Reading, Massachusetts|year = 1980|pages = 198-214|isbn = 0-201-13510-8
*cite book|last = Wall|first = H. S.|title = Analytic Theory of Continued Fractions|publisher = Chelsea Publishing Company|year = 1973|pages = 335-361|isbn = 0-8284-0207-8
(This is a reprint of the volume originally published by D. Van Nostrand Company, Inc., in 1948.)


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