In the
analytic theory of
continued fractions
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
, Euler's continued fraction formula is an identity connecting a certain very general
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
with an infinite
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension to the infinite case was immediately apparent. Today it is more fully appreciated as a useful tool in analytic attacks on the general
convergence problem for infinite continued fractions with complex elements.
The original formula
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 ...
derived the formula as
connecting a finite sum of products with a finite
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
.
:
The identity is easily established by
induction on ''n'', and is therefore applicable in the limit: if the expression on the left is extended to represent a
convergent infinite series, the expression on the right can also be extended to represent a convergent infinite
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
.
This is written more compactly using
generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.
A ge ...
notation:
:
Euler's formula
If ''r''
''i'' are complex numbers and ''x'' is defined by
:
then this equality can be proved by induction
:
.
Here equality is to be understood as equivalence, in the sense that the 'th
convergent of each continued fraction is equal to the 'th partial sum of the series shown above. So if the series shown is convergent – or ''uniformly'' convergent, when the ''r''
''i'''s are functions of some complex variable ''z'' – then the continued fractions also converge, or converge uniformly.
[H. S. Wall, ''Analytic Theory of Continued Fractions'', D. Van Nostrand Company, Inc., 1948; reprinted (1973) by Chelsea Publishing Company , p. 17.]
Proof by induction
Theorem: Let
be a natural number. For
complex values
,
:
and for
complex values
,
Proof: We perform a double induction. For
, we have
:
and
:
Now suppose both statements are true for some
.
We have
where
by applying the induction hypothesis to
.
But if
implies
implies
, contradiction. Hence
:
completing that induction.
Note that for
,
:
if
, then both sides are zero.
Using
and
,
and applying the induction hypothesis to the values
,
:
completing the other induction.
As an example, the expression
can be rearranged into a continued fraction.
:
This can be applied to a sequence of any length, and will therefore also apply in the infinite case.
Examples
The exponential function
The exponential function ''e''
''x'' is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
with a power series expansion that converges uniformly on every bounded domain in the complex plane.
:
The application of Euler's continued fraction formula is straightforward:
:
Applying an
equivalence transformation that consists of clearing the fractions this example is simplified to
:
and we can be certain that this continued fraction converges uniformly on every bounded domain in the complex plane because it is equivalent to the power series for ''e''
''x''.
The natural logarithm
The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
for the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are use ...
of the natural logarithm in the neighborhood of ''x'' = 1 is well known:
:
This series converges when , ''x'', < 1 and can also be expressed as a sum of products:
[This series converges for , ''x'', < 1, by ]Abel's test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test &nd ...
(applied to the series for log(1 − ''x'')).
:
Applying Euler's continued fraction formula to this expression shows that
:
and using an equivalence transformation to clear all the fractions results in
:
This continued fraction converges when , ''x'', < 1 because it is equivalent to the series from which it was derived.
The trigonometric functions
The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
function converges over the entire complex plane and can be expressed as the sum of products.
:
Euler's continued fraction formula can then be applied
:
An equivalence transformation is used to clear the denominators:
:
The same
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
can be applied to the
cosine function:
:
:
The inverse trigonometric functions
The
inverse trigonometric functions
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
can be represented as continued fractions.
:
An equivalence transformation yields
:
The continued fraction for the
inverse tangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
is straightforward:
:
A continued fraction for π
We can use the previous example involving the inverse tangent to construct a continued fraction representation of
π. We note that
:
And setting ''x'' = ''1'' in the previous result, we obtain immediately
:
The hyperbolic functions
Recalling the relationship between the
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s and the trigonometric functions,
:
:
And that
the following continued fractions are easily derived from the ones above:
:
:
The inverse hyperbolic functions
The
inverse hyperbolic functions
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.
For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. T ...
are related to the inverse trigonometric functions similar to how the hyperbolic functions are related to the trigonometric functions,
:
:
And these continued fractions are easily derived:
:
:
See also
*
Gauss's continued fraction
*
Engel expansion The Engel expansion of a positive real number ''x'' is the unique non-decreasing sequence of positive integers \ such that
:x=\frac+\frac+\frac+\cdots = \frac\left(1+\frac\left(1+\frac\left(1+\cdots\right)\right)\right)
For instance, Euler's cons ...
*
List of topics named after Leonhard Euler
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler inclu ...
References
{{Leonhard Euler
Continued fractions
Leonhard Euler