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complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
s, as well as some of the more complicated transcendental functions.


History

Lambert Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II *Lambert, Margrave of Tuscany ( fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name ...
published several examples of continued fractions in this form in 1768, and both
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaCarl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
who utilized the algebra described in the next section to deduce the general form of this continued fraction, in 1813. Although Gauss gave the form of this continued fraction, he did not give a proof of its convergence properties. Bernhard Riemann and L.W. Thomé 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.


Derivation

Let f_0, f_1, f_2, \dots be a sequence of analytic functions so that :f_ - f_i = k_i\,z\,f_ for all i > 0, where each k_i is a constant. Then :\frac = 1 + k_i z \frac, \text \frac = \frac Setting g_i = f_i / f_, :g_i = \frac, So :g_1 = \frac = \cfrac = \cfrac = \cfrac = \cdots.\ Repeating this ad infinitum produces the continued fraction expression :\frac = \cfrac In Gauss's continued fraction, the functions f_i are hypergeometric functions of the form _0F_1, _1F_1, and _2F_1, and the equations f_ - f_i = k_i z f_ arise as identities between functions where the parameters differ by integer amounts. These identities can be proven in several ways, for example by expanding out the series and comparing coefficients, or by taking the derivative in several ways and eliminating it from the equations generated.


The series 0F1

The simplest case involves :\,_0F_1(a;z) = 1 + \fracz + \fracz^2 + \fracz^3 + \cdots. Starting with the identity :\,_0F_1(a-1;z)-\,_0F_1(a;z) = \frac\,_0F_1(a+1;z), we may take :f_i = _0F_1(a+i;z),\,k_i = \tfrac, giving :\frac = \cfrac or :\frac = \cfrac. This expansion converges 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).


The series 1F1

The next case involves :_1F_1(a;b;z) = 1 + \fracz + \fracz^2 + \fracz^3 + \cdots for which the two identities :\,_1F_1(a;b-1;z)-\,_1F_1(a+1;b;z) = \frac\,_1F_1(a+1;b+1;z) :\,_1F_1(a;b-1;z)-\,_1F_1(a;b;z) = \frac\,_1F_1(a+1;b+1;z) are used alternately. Let :f_0(z) = \,_1F_1(a;b;z), :f_1(z) = \,_1F_1(a+1;b+1;z), :f_2(z) = \,_1F_1(a+1;b+2;z), :f_3(z) = \,_1F_1(a+2;b+3;z), :f_4(z) = \,_1F_1(a+2;b+4;z), etc. This gives f_ - f_i = k_i z f_ where k_1=\tfrac, k_2=\tfrac, k_3=\tfrac, k_4=\tfrac, producing :\frac = \cfrac or :\frac = \cfrac Similarly :\frac = \cfrac or :\frac = \cfrac Since _1F_1(0;b;z)=1, setting ''a'' to 0 and replacing ''b'' + 1 with ''b'' in the first continued fraction gives a simplified special case: :_1F_1(1;b;z) = \cfrac


The series 2F1

The final case involves :_2F_1(a,b;c;z) = 1 + \fracz + \fracz^2 + \fracz^3 + \cdots.\, Again, two identities are used alternately. :\,_2F_1(a,b;c-1;z)-\,_2F_1(a+1,b;c;z) = \frac\,_2F_1(a+1,b+1;c+1;z), :\,_2F_1(a,b;c-1;z)-\,_2F_1(a,b+1;c;z) = \frac\,_2F_1(a+1,b+1;c+1;z). These are essentially the same identity with ''a'' and ''b'' interchanged. Let :f_0(z) = \,_2F_1(a,b;c;z), :f_1(z) = \,_2F_1(a+1,b;c+1;z), :f_2(z) = \,_2F_1(a+1,b+1;c+2;z), :f_3(z) = \,_2F_1(a+2,b+1;c+3;z), :f_4(z) = \,_2F_1(a+2,b+2;c+4;z), etc. This gives f_ - f_i = k_i z f_ where k_1=\tfrac, k_2=\tfrac, k_3=\tfrac, k_4=\tfrac, producing :\frac = \cfrac or :\frac = \cfrac Since _2F_1(0,b;c;z)=1, setting ''a'' to 0 and replacing ''c'' + 1 with ''c'' gives a simplified special case of the continued fraction: :_2F_1(1,b;c;z) = \cfrac


Convergence properties

In this section, the cases where one or more of the parameters is a negative integer are excluded, since in these cases either the hypergeometric series are undefined or that they are polynomials so the continued fraction terminates. Other trivial exceptions are excluded as well. In the cases _0F_1 and _1F_1, the series converge everywhere so the fraction on the left hand side is a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...
. The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function. In the case _2F_1, the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle. The continued fractions on the right hand side will converge to the function everywhere inside this circle. Outside the circle, the continued fraction represents the
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
of the function to the complex plane with the positive real axis, from to the point at infinity removed. In most cases is a branch point and the line from to positive infinity is a branch cut for this function. The continued fraction converges to a meromorphic function on this domain, and it converges uniformly on any closed and bounded subset of this domain that does not contain any poles.


Applications


The series 0''F''1

We have :\cosh(z) = \,_0F_1(;), :\sinh(z) = z\,_0F_1(;), so :\tanh(z) = \frac = \cfrac = \cfrac. This particular expansion is known as Lambert's continued fraction and dates back to 1768. It easily follows that :\tan(z) = \cfrac. The expansion of tanh can be used to prove that ''e''''n'' is irrational for every integer ''n'' (which is alas not enough to prove that ''e'' is
transcendental Transcendence, transcendent, or transcendental may refer to: Mathematics * Transcendental number, a number that is not the root of any polynomial with rational coefficients * Algebraic element or transcendental element, an element of a field exten ...
). The expansion of tan was used by both Lambert and Legendre to prove that π is irrational. The
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
J_\nu can be written :J_\nu(z) = \frac\,_0F_1(\nu+1;-\frac), from which it follows :\frac=\cfrac. These formulas are also valid for every complex ''z''.


The series 1F1

Since e^z = _1F_1(1;1;z), 1/e^z = e^ :e^z = \cfrac :e^z = 1 + \cfrac. With some manipulation, this can be used to prove the simple continued fraction representation of '' e'', :e=2+\cfrac The
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...
erf (''z''), given by : \operatorname(z) = \frac\int_0^z e^ \, dt, can also be computed in terms of Kummer's hypergeometric function: : \operatorname(z) = \frac e^ \,_1F_1(1;;z^2). By applying the continued fraction of Gauss, a useful expansion valid for every complex number ''z'' can be obtained: : \frac e^ \operatorname(z) = \cfrac. 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 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
.


The series 2F1

From :(1-z)^=_1F_0(b;;z)=\,_2F_1(1,b;1;z), :(1-z)^ = \cfrac It is easily shown that the Taylor series expansion of arctan ''z'' in a neighborhood of zero is given by : \arctan z = zF(,1;;-z^2). The continued fraction of Gauss can be applied to this identity, yielding the expansion : \arctan z = \cfrac , 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. 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 = \cfrac = 1 - \frac + \frac - \frac \pm \cdots 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 natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, the arcsin function, and the generalized binomial series.


Notes


References

* *
(This is a reprint of the volume originally published by D. Van Nostrand Company, Inc., in 1948.) * {{MathWorld, title=Gauss's Continued Fraction, urlname=GausssContinuedFraction Continued fractions