Euler's continued fraction formula

In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. 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.[1] 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 derived the formula as an identity connecting a finite sum of products with a finite continued fraction.

\[ a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \cfrac{a_0}{1 - \cfrac{a_1}{1 + a_1 - \cfrac{a_2}{1 + a_2 - \cfrac{\ddots}{\ddots \cfrac{a_{n-1}}{1 + a_{n-1} - \cfrac{a_n}{1 + a_n}}}}}}\, \]

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.
Euler's formula in modern notation

If

\[ x = \cfrac{1}{1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}}\, \]

is a continued fraction with complex elements and none of the denominators B_i are zero,[2] a sequence of ratios {ri} can be defined by

\[ r_i = -\frac{a_{i+1}B_{i-1}}{B_{i+1}}.\, \]

For x and ri so defined, these equalities can be proved by induction.

\[ x = \cfrac{1}{1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}} = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\, \]

\[ x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = 1 + \sum_{i=1}^\infty \left( \prod_{j=1}^i r_j \right)\, \]

Here equality is to be understood as equivalence, in the sense that the nth convergent of each continued fraction is equal to the nth partial sum of the series shown above. So if the series shown is convergent – or uniformly convergent, when the ais and bis are functions of some complex variable z – then the continued fractions also converge, or converge uniformly.[3]
Examples
The exponential function

The exponential function ez is an entire function with a power series expansion that converges uniformly on every bounded domain in the complex plane.

\[ e^z = 1 + \sum_{n=1}^\infty \frac{z^n}{n!} = 1 + \sum_{n=1}^\infty \left(\prod_{j=1}^n \frac{z}{j}\right)\, \]

The application of Euler's continued fraction formula is straightforward:

\[ e^z = \cfrac{1}{1 - \cfrac{z}{1 + z - \cfrac{\frac{1}{2}z}{1 + \frac{1}{2}z - \cfrac{\frac{1}{3}z} {1 + \frac{1}{3}z - \cfrac{\frac{1}{4}z}{1 + \frac{1}{4}z - \ddots}}}}}.\, \]

Applying an equivalence transformation that consists of clearing the fractions this example is simplified to

\[ e^z = \cfrac{1}{1 - \cfrac{z}{1 + z - \cfrac{z}{2 + z - \cfrac{2z}{3 + z - \cfrac{3z}{4 + z - \ddots}}}}}\, \]

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 ez.
The natural logarithm

The Taylor series for the principal branch of the natural logarithm in the neighborhood of z = 1 is well known. Recognizing that log(a/b) = log(a) - log(b), the following series is easily derived:

\[ \log \frac{1+z}{1-z} = 2\left(z + \frac{z^3}{3} + \frac{z^5}{5} + \cdots\right) = 2\sum_{n=0}^\infty\] \frac{z^{2n+1}}{2n+1}.\,

This series converges when |z| < 1 and can also be expressed as a sum of products:[4] \]

\[ \begin{align} \log \frac{1+z}{1-z} & = 2z \left[1 + \frac{z^2}{3} + \frac{z^4}{5} + \cdots\right] \\[8pt] & = 2z \left[1 + \frac{z^2}{3} + \left(\frac{z^2}{3}\right)\frac{z^2}{5/3} + \left(\frac{z^2}{3}\right)\left(\frac{z^2}{5/3}\right)\frac{z^2}{7/5} + \cdots\right] \end{align} \]

Applying Euler's continued fraction formula to this expression shows that

\[ \log \frac{1+z}{1-z} = \cfrac{2z}{1 - \cfrac{\frac{1}{3}z^2}{1 + \frac{1}{3}z^2 - \cfrac{\frac{3}{5}z^2}{1 + \frac{3}{5}z^2 - \cfrac{\frac{5}{7}z^2}{1 + \frac{5}{7}z^2 - \cfrac{\frac{7}{9}z^2}{1 + \frac{7}{9}z^2 - \ddots}}}}}\, \]

and using an equivalence transformation to clear all the fractions results in

\[ \log \frac{1+z}{1-z} = \cfrac{2z}{1 - \cfrac{z^2}{z^2 + 3 - \cfrac{(3z)^2}{3z^2 + 5 - \cfrac{(5z)^2}{5z^2 + 7 - \cfrac{(7z)^2}{7z^2 + 9 - \ddots}}}}}.\, \]

This continued fraction converges when |z| < 1 because it is equivalent to the series from which it was derived.[4]
A continued fraction for π

We can use the previous example involving the principal branch of the natural logarithm function to construct a continued fraction representation of π. First we note that

\[ \frac{1+i}{1-i} = i \quad\Rightarrow\quad \log\frac{1+i}{1-i} = \frac{i\pi}{2}.\, \]

Setting z = i in the previous result, and remembering that i² = −1, we obtain immediately

\[ \pi = \cfrac{4}{1 + \cfrac{1^2}{2 + \cfrac{3^2}{2 + \cfrac{5^2}{2 + \cfrac{7^2}{2 + \ddots}}}}}.\, \]

See also

List of topics named after Leonhard Euler

Notes

^ 1748 Leonhard Euler, Introductio in analysin infinitorum, Vol. I, Chapter 18.
^ These denominators B_i are determined by the fundamental recurrence formulas.
^ (Wall, 1948, p. 17)
^ a b This series converges for |z| = 1, except when z = ±1, by Abel's test (applied to the series for log(1 − z)).

References

H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted (1973) by Chelsea Publishing Company ISBN 0-8284-0207-8.