For many, this may be a first introduction to continued fractions since this subject, if it is taught at all, is restricted to a single chapter in a number theory text. It turns out, however, that fractions of this form, called continued fractions, provide much insight into many mathematical problems, particularly into the nature of numbers. Recently i have had some misgivings about using the other listed methods to compute continued fractions. Thus it is natural to look at the continued fraction expansion. The most common type of continued fraction is that of continued fractions for real numbers. Our derivation of the gauss continued fraction follows the lines of sec tion 2.
Lecture 1 we describe the farey tessellation fand give a very quick introduction to the basic facts we need from hyperbolic geometry, using the upper half plane model. On the use of continued fractions for stream ciphers. I dreamt that somehow i had come, to dwell in topsyturveydom. Continued fractions are one of many ways of representing real numbers.
We also state without proof some further properties of continued fractions and provide a brief overview of some facts in this connection. Such a fraction is called a terminating continued fraction. Continued fractions and pell equations this is not intended to be a complete set of notes on the subject. In 3, euler analyzed the ricatti equation to prove that the. The aim of this series is to apply continued fractions in a number of contexts, most notably, to the solution of pells equation, x2 dy2 1 where d is squarefree, and as a corollary, to. In this reu paper, i will use some important characteristics of continued fractions to give the complete set of solutions to pells equation. The nineteenth century can probably be described as the golden age of continued fractions. The derivative of a finite continued fraction jussi malilay received 4 may 20 abstract the derivative of a nite continued fraction of a complex variable is derived by presenting the continued fraction as a component of a nite composition of c 2. Hopefully this site will inspire others to study continued fractions in greater detail. If the simple continued fraction does not have 0 as its rst number, then shift all the numbers to the right and place 0 as the rst entry. Chapter 47 the topsyturvy world of continued fractions online the other night, from cares exempt, i sleptand what dyou think i dreamt. Notes edited by william adkins a short proof of the simple continued fraction expansion of e henry cohn 1.
We extend the formulas for the first and the second logarithmic derivatives of bs to the case of ys,r. Derivatives of continued fractions with applications to hypergeometric functions. We then briefly discuss the connections between the deduced formula and partial derivatives with respect to the elements ak. The first derivative of ramanujans cubic continued fraction. Chapter 17 continued fractions trinity college, dublin.
Continued fractions can contain a finite or infinite number of fractions. Book i, all without taking a single derivative or integral. Pdf continued fraction expansion approaches to discretizing. If the simple continued fraction has a 0 as its rst number, then remove the 0. Bowman2 1department of statistics, university of georgia athens, georgia 30602, u. Pdf the first derivative of ramanujans cubic continued fraction. Derivatives of continued fractions with applications to. Pdf continued fraction calculation of spherical bessel. This one is a little trickier to remember, but luckily it comes with its own song.
They have some interesting connections with a jigsawpuzzle problem about splitting a rectangle into squares and also with one of the oldest algorithms known to greek mathematicians of 300 bc euclids algorithm for computing the greatest divisor common to two numbers gcd. Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. Period of the continued fraction of v n marius beceanu february 5, 2003 abstract this paper seeks to recapitulate the known facts about the length of the period of the continued fraction expansion of v n as a function of n and to make a few possibly original contributions. Displayed in their full laddered form, they look like this. The derivative of a finite continued fraction rims, kyoto. Worksheet number thirteen continued fractions we are all familiar with decimal numbers, and know that a repeating decimal gives a rational number, while one that never repeats and goes on forever represents an irrational number. You can find more details and proofs either in texts or on the web. The 25th entry in chapter 12 of notebook ii is a continued fraction involving a quotient of gamma. The power rule fraction examples derivatives calculus. It is well known that one can obtain explicit continued fraction expansions of ez for various. Using continued fraction convergents to find a formula in terms of derivatives. They are examples of continued fractions, which are used to get best approximations to an irrational number for a given upper bound on the denominator, socalled diophantine approximation. How to find continued fraction of pi stack exchange.
Continued fractions, derivatives, hypergeometric functions. Walters, alternative derivation of some regular continued frac tions. Although we do not limit ourselves to this case in the course, it will be used. It was a time in which the subject was known to every mathematician. Lets use the pell equation x2 7y2 1, as our example. Continued fractions and pells equation seung hyun yang abstract. Mathematical shorthand notation for simple continued fractions is a list of just the integer parts, ignoring the numerators which are always 1.
A short proof of the simple continued fraction expansion of. Basic definitions and facts a continued fraction is given by two sequences of numbers bnn. The actual term continued fraction has been around for 350 years since it was first seen in john wallis arithmetica infinitorum in 1653. From now on, unless the contrary is stated, the words continued fraction will imply that we are dealing with a finite simple continued fraction. I have established a result concerning the average period length.
Continued fractions were studied by the great mathematicians of the seventeenth and eighteenth centuries and are a subject of active investigation today. As claude brezinski writes in history of continued fractions and padre approximations, the nineteenth century can be said to be popular period for continued fractions. Far east journal of applied mathematics volume 35, number 3, 2009, pages 363371 the derivative of a continued fraction l. A continued fraction is a fraction in which the numerators and denominators may contain continued fractions. The derivative of a finite continued fraction of a complex variable is derived by presenting the continued fraction as a component of a finite composition of c c linear fractional transformations of analytic functions. This video shows you how to find the derivative of a fraction using the power rule, chain rule, and quotient rule. Pdf continued fractions and factoring researchgate.
C linear fractional transformations of analytic functions. Continued fractions are, in some ways, more mathematically natural representations of a real number than other representations such as decimal representations, and they have several desirable properties. Using continued fraction convergents to find a formula in. The following formula for a sequence xnn1 is called the walliseuler recurrence formula. In this talk we introduce continued fractions, prove their basic properties and apply these properties to solve a practical problem. Proceedings of the roman number theory association volume 2, number 1, march 2017, pages 6181 michel waldschmidt continued fractions. Continued fraction expansions of values of the exponential function.
Feb 02, 2016 in this video, i work out an example of taking derivatives involving fractions not using the quotient rule. There is another way to represent real numbers called continued fractions. Continued fraction expansion approaches to discretizing. A note on partial derivatives of continued fractions. Derivatives of continued fractions with applications to hypergeometric functions haakon waadeland department of mathematics, university of trondheim a vii, n7055 dragvoll, norway received 10 march 1986 keywords. Continued fraction calculation of spherical bessel functions article pdf available in computers in physics 44. Thus the difference between successive convergents approaches zero and alternates in sign, so a continued fraction always converges to a real number. We begin with a diversion into the topic of continued fractions. Department of mathematics, university of trondheim a. Alternative derivation of some regular continued fractions. The continued fraction representation for a rational number is finite and only rational numbers have finite representations. Were upgrading the acm dl, and would like your input.
Journal of computational and applied mathematics 19 1987 161169 161 northholland derivatives of continued fractions with applications to hypergeometric functions haakon waadeland department of mathematics, university of trondheim avh, n7055 dragvoll, norway received 10 march 1986 keywords. Continued fractions and the euclidean algorithm lecture notes prepared for math 326, spring 1997 department of mathematics and statistics university at albany. The quotient rule now, we want to be able to take the derivative of a fraction like f g, where f and g are two functions. Connections to previous work and possible applications of the deduced formula are briefly discussed. It seems very inconvenient to be required to have a very good decimal approximation of your number before computing the convergents you want. Jun 23, 20 in this video i go over a couple of example questions finding the derivative of functions with fractions in them using the power rule. Continued fractions are just another way of writing fractions. This continued fraction is a generalization of the brounckers continued fraction bs. The implementation uses a maple code for derivatives.
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