In
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 Lagrangia[Carl 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 ...](_blank)
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
be a sequence of analytic functions so that
:
for all
, where each
is a constant.
Then
:
Setting
:
So
:
Repeating this ad infinitum produces the continued fraction expression
:
In Gauss's continued fraction, the functions
are hypergeometric functions of the form
,
, and
, and the equations
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
:
Starting with the identity
:
we may take
:
giving
:
or
:
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
:
for which the two identities
:
:
are used alternately.
Let
:
:
:
:
:
etc.
This gives
where
, producing
:
or
:
Similarly
:
or
:
Since
, setting ''a'' to 0 and replacing ''b'' + 1 with ''b'' in the first continued fraction gives a simplified special case:
:
The series 2F1
The final case involves
:
Again, two identities are used alternately.
:
:
These are essentially the same identity with ''a'' and ''b'' interchanged.
Let
:
:
:
:
:
etc.
This gives
where
, producing
:
or
:
Since
, setting ''a'' to 0 and replacing ''c'' + 1 with ''c'' gives a simplified special case of the continued fraction:
:
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
and
, 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
, 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
:
:
so
:
This particular expansion is known as Lambert's continued fraction and dates back to 1768.
It easily follows that
:
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 ...
can be written
:
from which it follows
:
These formulas are also valid for every complex ''z''.
The series 1F1
Since
,
:
:
With some manipulation, this can be used to prove the simple continued fraction representation of
''
e'',
:
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
:
can also be computed in terms of Kummer's hypergeometric function:
:
By applying the continued fraction of Gauss, a useful expansion valid for every complex number ''z'' can be obtained:
:
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
:
:
It is easily shown that the Taylor series expansion of
arctan ''z'' in a neighborhood of zero is given by
:
The continued fraction of Gauss can be applied to this identity, yielding the expansion
:
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
:
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