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Plouffe's Constants

Plouffe's constants are numbers arising in summations of series related to where is a trigonometric function. Define the Iverson bracket function(1)Now define through(2)(3)then(4)(5)(6)(OEIS A086201).For(7)(8)the sum is (amazingly) given by(9)(10)(11)(OEIS A086202), where denotes the XOR of binary digits (Chowdhury 2001a; Finch 2003, p. 432). A related sum is given by(12)(13)(14)(OEIS A111953), where again denotes the XOR of binary digits (Chowdhury 2001b; Finch 2005, p. 20).Letting(15)(16)then(17)(18)(OEIS A049541).Plouffe asked if the above processes could be "inverted." He considered(19)(20)giving(21)(22)and(23)(24)giving(25)(26)and(27)(28)giving(29)(30)(31)(OEIS A086203), where the identity was conjecturedby Plouffe and proved by Borwein and Girgensohn (1995). is sometimes known as Plouffe's constant (Plouffe 1997), although this angle had arisen in the geometry of the..

Feigenbaum Constant Approximations

A curious approximation to the Feigenbaum constant is given by(1)where is Gelfond's constant, which is good to 6 digits to the right of the decimal point.M. Trott (pers. comm., May 6, 2008) noted(2)where is Gauss's constant, which is good to 4 decimal digits, and(3)where is the tetranacci constant, which is good to 3 decimal digits.A strange approximation good to five digits is given by the solution to(4)which is(5)where is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).(6)gives to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).M. Hudson (pers. comm., Nov. 20, 2004) gave(7)(8)(9)which are good to 17, 13, and 9 digits respectively.Stoschek gave the strange approximation(10)which is good to 9 digits.R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations(11)(12)(13)(14)(15)(16)where e is the base of the natural logarithm and..

Natural Logarithm

The natural logarithm is the logarithm having base e, where(1)This function can be defined(2)for .This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(3)The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .Common and natural logarithms can be expressed in terms of each other as(4)(5)The natural logarithm is especially useful in calculusbecause its derivative is given by the simple equation(6)whereas logarithms in other bases have the more complicated..

Natural Logarithm of 2

The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..

Pi Formulas

There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..

Riemann Zeta Function

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..

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