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 ex. The natural logarithm The Taylor series for the principal branch 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: 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 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 The inverse trigonometric functions can be represented as continued fractions. An equivalence transformation yields The continued fraction for the inverse tangent 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 functions 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 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 List of topics named after Leonhard Euler References Continued fractions Leonhard Euler