Thomas Osler

• Ph D Courant Institute, NYU 1970
• Professor at Rowan University

In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Eule...

100.36 Another look at higher vertical motion - Volume 100 Issue 549 - Thomas J. Osler

The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations (1) and (2) (Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a 0 , the smaller has perimeter b 0 . (Arch...

We present three elementary approximate formulas for the period of a pendulum which starts at rest from a large angle of displacement. The first of these formulas is known, but the other two may be new. These three formulas result from taking the first three partial products of a new infinite product of nested radicals for the complete elliptic int...

We give an iterative algorithm that converges to the lemniscate constant L. This algorithm resembles the famous Archimedean algorithm for pi. The derivation is based on the recently discovered product of nested radicals for 2/L by Aaron Levin. Levin's product closely resembles Vieta's historic product for 2/pi.

The arithmetic-geometric mean of two positive numbers a and b (AGM(a,b)) is the common limit of two sequences generated by an iterative process. This has proven to be an important device for calculating numbers and function in recent years. In this paper, we derive an infinite product representation for the AGM. The factors of this product are nest...

Short and fuzzy derivations of five remarkable formulas for primes - Volume 99 Issue 545 - Thomas J. Osler

We introduce four methods of dealing with divergent series that have been shown to be of value in analysis. These are Cesaro summation, Abel summation, Borel summation, and asymptotic series. We mention how Euler used divergent series to discover the famous functional equation for the zeta function. This is an expository paper to gently introduce t...

98.35 A novel approach to finding ∫ cosec x dx - Volume 98 Issue 543 - Walter Jacob, Thomas J. Osler

In this paper we show how to derive Vieta’s famous product of nested radicals for π [1] from the Archimedean iterative algorithm for π, [2, 3]. Only simple algebraic manipulations are needed. The Archimedean iterative algorithm for calculating π uses a method involving the two equations: and In using this algorithm, we start with a circle of diame...

98.23 Recursive formulas related to the Arithmetic–Geometric Mean - Volume 98 Issue 543 - Thomas J. Osler, Tirupathi R Chandrupatla

Vieta's product for 2π has factors that arc nested radicals. The Wallis product for 2π has factors that are rational numbers. Brouncker gave continued fractions for 4π. By summarizing some recently published results in this expository paper, we show that these seemingly unrelated results are connected. We give a general formula in which the product...

97.09 The devil's series – Did it fool Euler? - Volume 97 Issue 538 - Thomas J. Osler, Steven Donahue

97.08 Interesting bilateral series generalising a result of Euler - Volume 97 Issue 538 - Cory Wright, Thomas J. Osler

Euler discovered an interesting method for astronomers to calculate the eccentricity of a celestial body in an elliptical orbit. We describe the mathematics behind Euler's method and show how it can be used by astronomical observers.

96.53 Partial sums of series that cannot be an integer - Volume 96 Issue 537 - Thomas J. Osler

The forgotten continued fractions Lord Brouncker's continued fraction for π is (In this paper we will use the more convenient notation This fraction first appeared in the Arithmetica Infinitorum [1] by John Wallis. In this book, along with topics that lead to Newton's calculus, Wallis derives his famous product Wallis tells us that he showed this...

96.09 Euler's method of integration by parts - Volume 96 Issue 535 - Steven Donahue, Thomas J. Osler

We present four derivations of the closed form of the partial fractions expansion This interesting series is a generalization of the series made famous by Euler.

95.41 An interesting generalisation of the half-angle formulas to complex numbers - Volume 95 Issue 533 - Marcus Wright, Thomas J. Osler

The law of cosines for any triangle with sides of length a, b and c is c ² = a ² + b ² − 2ab cos γ , where γ is the angle opposite side c . We show that this law generalises nicely to any polygon as well as any polyhedron. The generalisation to a quadrilateral with sides a, b, c and d is and for a five sided polygon

Three of the oldest and most celebrated formulae for π are: The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenie...

In the paper [1], Euler was examining properties of the conic sections that could be shared by more general curves. Most of the paper is concerned with ‘oblique-angle diameters’, a concept that seems to have been familiar to his readers in the eighteenth century, but has been ignored today. In this paper we will explain this concept and, led by Eul...

A biarc consists of two circular arcs which are tangential where they meet. This paper presents a geometric view of piecewise circular curves with continuously varying tangents (1 C or 1 G curves). A smooth curve is divided into smaller segments by placing segment end points at various locations. In particular, these are placed at points of inflect...

In the year 1656 John Wallis published his Arithmetica Infinitorum , [1], in which he displayed many ideas that were to lead to the integral calculus of Newton. In this work we find the celebrated infinite product of Wallis which gives π , Earlier in 1593, Vieta [2] found another infinite product which gives π But, since Wallis does not mention i...

The beautiful infinite product of radicals due to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products: for N even, and for N odd. Here N is a positive integer, F N and L N are the Fibonacci and Lucas numbers, and is the golden section....

‘Lord Brouncker's continued fraction for π’ is a well-known result. In this article, we show that Brouncker found not only this one continued fraction, but an entire infinite sequence of related continued fractions for π. These were recorded in the Arithmetica Infinitorum by John Wallis, but appear to have been ignored and forgotten by modern mathe...

In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating series The ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by

Euler, in his paper E352, [3,7]. conjectured an equation which is equivalent to the functional equation for the zeta function. This work of Euler was forgotten, and one hundred years later Riemann would rediscover this functional equation and provide a rigorous proof. All modern proofs of the functional equation involve mathematical tools that were...

92.43 Euler’s little summation formula and special values of the zeta function - Volume 92 Issue 524 - Thomas J. Osler

Data from at least three observations of an asteroid or planet is required to calculate the true elliptical orbit. Using ecliptic longitudes from only two close observations, we try to compute a circle that approximates the elliptical orbit. Our calculations result in not one, but two circular orbits. In some cases, we can use physical arguments to...

We present two infinite products of nested radicals involving Fibonacci and Lucas numbers. These products resemble Vieta’s classical product of nested radicals for 2/π. A modern derivation of Vieta’s product involves trigonometric functions, while our product involves similar manipulations involving hyperbolic functions.

Vieta's famous product using factors that are nested radicals is the oldest infinite product as well as the first non-iterative method for finding Ï-. In this paper a simple geometric construction intimately related to this product is described. The construction provides the same approximations to Ï- as are given by partial products from Vieta's fo...

In some elementary courses, it is shown that square root of 2 is irrational. It is also shown that the roots like square root of 3, cube root of 2, etc., are irrational. Much less often, it is shown that the number "e," the base of the natural logarithm, is irrational, even though a proof is available that uses only elementary calculus. In this sho...

Euler gave a simple method for showing that ζ(2) = 1/1 + 1/2 + 1/3+ ··· = π/6. He generalized his method so as to find ζ(4), ζ(6), ζ(8), … . His computations became increasingly more complex as the arguments increased. In this note we show a different generalization of Euler's original method for finding ζ(2p) that is simpler to understand and is s...

The analysis of tautochrone problems involves the solution of integral equations. The paper shows how a reasonable assumption, based on experience with simple harmonic motion, allows one to greatly simplify such problems. Proposed solutions involve only mathematics available to students from first year calculus.

Tom Osler (osler@rowan. edu) is a professor of mathematics at Rowan University (Glassboro, NJ 08028). He received his Ph. D. from the Courant Institute at New York University in 1970 and is the author of seventy-five mathematical papers. In addition to teaching university mathematics for the past forty-four years, Tom has a passion for long distanc...

The two oldest representations for the number π are infinite product expansions. The first, is due to Vieta in 1592. The second is Wallis's product dating from 1655:

Theon's ladder is an ancient algorithm for calculating rational approximations for . It features two columns of integers (called a ladder), in which the ratio of the two numbers in each row is an approximation to . It is remarkable for its simplicity. This algorithm can easily be generalized to find rational approximations to any square root. In th...

In rare cases, the use of advanced mathematical calculators can give incorrect results. One such error occurs with graphing calculators because the screen is not continuous, but a rectangular array of pixels. On some frequently used calculators, the graph of sin 80x looks like sin x. We also study other related examples.

It has been shown recently that the two oldest infinite product representations of π , Vieta's product of radicals, and Wallis's product of rationals, are both special cases of another infinite product called the Vieta-Wallis product (VWP). In this new product there is a parameter p. When p = 0, we get Wallis's product, and as p grows to infinity,...

Several formulae for the inradius of various types of triangles are derived. Properties of the inradius and trigonometric functions of the angles of Pythagorean and Heronian triangles are also presented. The entire presentation is elementary and suitable for classes in geometry, precalculus mathematics and number theory.

A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx) + p(x)v = q(x) by means of a substitution v = f(y). The student then creates a table of the new equations and de...

An intuitive derivation of Stirling's formula is presented, together with a modification that greatly improves its accuracy. The derivation is based on the closed form evaluation of the gamma function at an integer plus one-half. The modification is easily implemented on a hand-held calculator and often triples the number of significant digits calc...

Kepler's first law of planetary motion states that the orbits of planets are elliptical, with the sun at one focus. We present an unusual verification of this law for use in classes in mechanics. It has the advantages of resembling the simple verification of circular orbits, and stressing the importance of Kepler's equation.

In a recent paper by Flores and Osler, the authors investigated tautochrone curves in the xy-plane under an arbitrary potential V(y). In this paper we imagine that the xy-plane of the tautochrone curve is rotating about the y-axis with constant angular momentum. We find the differential equation of the tautochrone curves. While this differential eq...

triangle. Figure 1 shows the incircle for a triangle. It is easy to see that the center of the incircle (incenter) is at the point where the angle bisectors of the triangle meet. In this note we refer to a right triangle in which all three sides are relatively prime as a Pythagorean triangle. The two most familiar are the 3, 4, 5 and the 5, 12, 13...

Most readers will not have encountered a derivative of "order 1/2" before, because almost none of the familiar text books mention it. Yet the notion was discussed briefly as early as the eighteenth century by Leibniz. Other giants of the past including L'Hospital, Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with...

) While (1) and (2) seem unrelated, the above expression (3) shows that they are both special cases of a more general "double product". The first product in (3) consists of the first p factors of Vieta's original infinite product (1). The second product in (3) is a Wallis-like product. We say this because the case where p = 0 gives us the original...

The classical tautochrone problem involves motion along curves caused by the special potential V y mgy () = . We use fractional derivatives to find tautochrone curves under arbitrary potentials V y () . We generalize these further to potentials that are functions of two variables V x y (,) . An appendix gives intuitive motivation for the fractional...

Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file. (AS...

The fractional derivative operator is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representations which have been proposed for this concept are reviewed in this paper. In particular, those representations which appear to be of greatest interest for...

In this paper, various representations of fractional differentiation are explored, and a definition using Pochhammer contour integrals emerges as deserving special emphasis. The analyticity of Dαzpf(z) and Dαzpln z f(z) is investigated with reference to the three variables z, α, and p. The validity of the operation DβDα=Dα+α is studied. An improvem...