 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 importantelementary function s, as well as some of the more complicatedtranscendental function s.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. 134138.]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. 400406. (Posthumous fragment).] and Thomé [L. W. Thomé (1867), "Über die Kettenbrüchentwicklung des Gaussen quotienten …," "Jour. für Math." vol. 67 pp. 299309.] obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, byEdward 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. 118.]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.]

STYLE="VERTICALALIGN: TOP"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)]
:$frac\{F(a,b+1;c+1;z)\}\{F(a,b;c;z)\}\; =\; cfrac\{1\}\{1\; \; cfrac\{frac\{a(cb)\}\{c(c+1)\}z\}\{1\; \; cfrac\{frac\{(b+1)(ca+1)\}\{(c+1)(c+2)\}z\}\{1\; \; cfrac\{frac\{(a+1)(cb+1)\}\{(c+2)(c+3)\}z\}\{1\; \; cfrac\{frac\{(b+2)(ca+2)\}\{(c+3)(c+4)\}z\}\{1\; \; cfrac\{frac\{(a+2)(cb+2)\}\{(c+4)(c+5)\}z\}\{1\; \; ddots.$
Convergence properties
Provided only that mathc 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 theanalytic continuation of the function mathf(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 matha is zero or a negative integer, or if mathb 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 mathz. If matha is neither zero nor a negative integer, and mathb 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:
:$F(a,1;c;z)\; =\; cfrac\{1\}\{1\; \; cfrac\{frac\{a\}\{c\}z\}\{1\; \; cfrac\{frac\{ca\}\{c(c+1)\}z\}\{1\; \; cfrac\{frac\{c(a+1)\}\{(c+1)(c+2)\}z\}\{1\; \; cfrac\{frac\{2(ca+1)\}\{(c+2)(c+3)\}z\}\{1\; \; cfrac\{frac\{(c+1)(a+2)\}\{(c+3)(c+4)\}z\}\{1\; \; cfrac\{frac\{3(ca+2)\}\{(c+4)(c+5)\}z\}\{1\; \; ddots\}.$
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_{1}F_{1} is sometimes written as mathM(a; c; z), or (in both Wall (1973) and Jones & Thron (1980)) as mathΦ(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
:$frac\{,\_1F\_1(a+1;c+1;z)\}\{,\_1F\_1(a;c;z)\}\; =\; cfrac\{1\}\{1\; cfrac\{frac\{ca\}\{c(c+1)\}z\}\{1\; +\; cfrac\{frac\{a+1\}\{(c+1)(c+2)\}z\}\{1\; \; cfrac\{frac\{ca+1\}\{(c+2)(c+3)\}z\}\{1\; +\; cfrac\{frac\{a+2\}\{(c+3)(c+4)\}z\}\{1\; \; cfrac\{frac\{ca+2\}\{(c+4)(c+5)\}z\}\{1\; +\; \; ddots.$
Since math_{1}F_{1}(a; c; z) is an
entire function of mathz (provided that mathc ≠ 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_{1}F_{1}(0; c; z) is equal to unity, this formula can also be simplified. Substituting matha = 0 in the preceding formula, replacing mathc + 1 by mathc, 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 point s 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 integral s, for theDawson function , and for theincomplete gamma function . A simpler version of the argument yields two useful continued fraction expansions of theexponential function . [See the example in the articlePadé table for the expansions of "e^{z}" 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 booklast = Jonesfirst = William B.coauthors = Thron, W. J.title = Continued Fractions: Theory and Applicationspublisher = AddisonWesley Publishing Companylocation = Reading, Massachusettsyear = 1980pages = 198214isbn = 0201135108
*cite booklast = Wallfirst = H. S.title = Analytic Theory of Continued Fractionspublisher = Chelsea Publishing Companyyear = 1973pages = 335361isbn = 0828402078
(This is a reprint of the volume originally published by D. Van Nostrand Company, Inc., in 1948.)
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