derived the formula as connecting a finite sum of products with a finite continued fraction. 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
If ri 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 n'th convergent of each continued fraction is equal to the n'th partial sum of the series shown above. So if the series shown is convergent - or uniformly convergent, when the ri's are functions of some complex variablez - then the continued fractions also converge, or converge uniformly.
Proof
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.
The exponential function ez is an entire function 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 ez.
The Taylor series for the principal branch of the natural logarithm in the neighborhood of z = 1 is well known: This series converges when |z| < 1 and can also be expressed as a sum of products: 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 |z| < 1 because it is equivalent to the series from which it was derived.
The Taylor series of the sine 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 can be applied to the cosine function:
The inverse trigonometric functions can be represented as continued fractions. An equivalence transformation yields The continued fraction for the inverse tangent is straightforward:
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
Recalling the relationship between the hyperbolic functions and the trigonometric functions, And that the following continued fractions are easily derived from the ones above:
The inverse hyperbolic functions 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:
