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((x),(y) = {239, 13 2} is a solution to the Pell equation x 2 − 2 y 2 = −1.) Formulae of this kind are known as Machin-like formulae . Machin's particular formula was used well into the computer era for calculating record numbers of digits of π , [ 39 ] but more recently other similar formulae have been used as well.
mathematical constant π. 3.14159 26535 89793 23846 26433... The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π.
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [ a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted ...
Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this ...
Rogers–Ramanujan identities. In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913.
22 7 is a widely used Diophantine approximation of π. It is a convergent in the simple continued fraction expansion of π. It is greater than π, as can be readily seen in the decimal expansions of these values: The approximation has been known since antiquity. Archimedes wrote the first known proof that 22 7 is an overestimate in ...
The Bogacki–Shampine method is a method for the numerical solution of ordinary differential equations, that was proposed by Przemysław Bogacki and Lawrence F. Shampine in 1989 (Bogacki & Shampine 1989). The Bogacki–Shampine method is a Runge–Kutta method of order three with four stages with the First Same As Last (FSAL) property, so that ...
For arbitrary stencil points and any derivative of order up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] where is the Kronecker delta, equal to one if , and zero otherwise. Example, for , order of differentiation :
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