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Continued fractions are most conveniently applied to solve the general quadratic equation expressed in the form of a monic polynomial x 2 + b x + c = 0 {\displaystyle x^{2}+bx+c=0} which can always be obtained by dividing the original equation by its leading coefficient .
In the first of these equations the ratio tends toward A n / B n as z tends toward zero. In the second, the ratio tends toward A n / B n as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents A n / B n are eventually arbitrarily close ...
Euler 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 ...
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
Partial fractions (Heaviside's method) ... The Atangana–Baleanu fractional integral of a continuous function is ... when solving differential equations using Caputo ...
In mathematics, regular continued fractions play an important role in representing real numbers, and have a rich general theory touching on a variety of topics in number theory. Moreover, generalized continued fractions have important and interesting applications in complex analysis
Inverse function integration (a formula that expresses the antiderivative of the inverse f −1 of an invertible and continuous function f, in terms of f −1 and the antiderivative of f). The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials) The Risch algorithm
the sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition.