# 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) |

and volume enclosed are

(3) | |||

(4) |

tangents of unit fractions is Machin's

formula

(5) |

as well as thousands of other similar formulas having more terms.

Gregory and Leibniz found

(6) | |||

(7) |

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 to

(8) |

zeta function (Vardi 1991, pp. 157-158; Flajolet and Vardi 1996), so that

the error after terms is .

An infinite sum series to Abraham Sharp (ca. 1717) is given by

(9) |

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) |

In 1666, Newton used a geometric construction to derive the formula

(18) | |||

(19) |

*et*

al.1989; Borwein and Bailey 2003, pp. 105-106). The coefficients can be

al.

found from the integral

(20) | |||

(21) |

0, obtaining

(22) |

transformation gives

(23) | |||

(24) | |||

(25) |

*et al.*1972, Item 120).

This corresponds to plugging into

the power series for the hypergeometric

function ,

(26) |

gives 2 bits/term,

(27) |

(28) |

ratio. Gosper also obtained

(29) |

and Wagon (1995; Borwein and Bailey 2003, pp. 141-142).

More amazingly still, a closed form expression giving a digit-extraction algorithm which produces digits of (or ) in base-16

was discovered by Bailey

*et al.*(Bailey

*et al.*1997, Adamchik and Wagon

1997),

(30) |

*et al.*1999)

and is equivalent to

(31) |

few independent formulas of which are

(32) | |||

(33) | |||

(34) | |||

(35) | |||

(36) | |||

(37) |

independent formulas of which are

(38) | |||

(39) | |||

(40) | |||

(41) | |||

(42) | |||

(43) | |||

(44) | |||

(45) | |||

(46) | |||

(47) | |||

(48) |

formula

(49) |

(50) |

Girgensohn 2004, p. 3; Boros and Moll 2004, p. 125; Lucas 2005; Borwein

*et al.*

2007, p. 14). This integral was known by K. Mahler in the mid-1960s

and appears in an exam at the University of Sydney in November 1960 (Borwein, Bailey,

and Girgensohn, p. 3). Beukers (2000) and Boros and Moll (2004, p. 126)

state that it is not clear if these exists a natural choice of rational polynomial

whose integral between 0 and 1 produces , where

333/106 is the next convergent. However, an integral exists for the

*fourth*

convergent, namely

(51) |

*et al.*2007, p. 219). In fact, Lucas (2005) gives

a few other such integrals.

Backhouse (1995) used the identity

(52) | |||

(53) | |||

(54) |

to generate a number of formulas for . In particular,

if , then (Lucas 2005).

A similar formula was subsequently discovered by Ferguson, leading to a

two-dimensional lattice of such formulas which can be generated by these

two formulas given by

(55) |

formula as the special case .

An even more general identity due to Wagon is given by

(56) |

axis, as illustrated above.

A perhaps even stranger general class of identities is given by

(57) |

symbol (B. Cloitre, pers. comm., Jan. 23, 2005). Even more amazingly,

there is a closely analogous formula for the natural

logarithm of 2.

Following the discovery of the base-16 digit BBP formula and related formulas, similar formulas in other bases were investigated. Borwein,

Bailey, and Girgensohn (2004) have recently shown that has no Machin-type

BBP arctangent formula that is not binary, although this does not rule out a completely

different scheme for digit-extraction algorithms

in other bases.

S. Plouffe has devised an algorithm to compute the th digit

of in any base in steps.

A slew of additional identities due to Ramanujan, Catalan, and Newton

are given by Castellanos (1988ab, pp. 86-88), including several

involving sums of Fibonacci

numbers. Ramanujan found

(58) |

Plouffe (2006) found the beautiful formula

(59) |

An interesting infinite product formula due to Euler which relates and the th prime is

(60) | |||

(61) |

A method similar to Archimedes' can be used to estimate by starting with

an -gon and then relating the area

of subsequent -gons. Let be the angle

from the center of one of the polygon's segments,

(62) |

(63) |

Vieta (1593) was the first to give an exact expression for by taking in the above expression, giving

(64) |

radicals,

(65) |

A related formula is given by

(66) |

(67) |

(68) |

comm., April 27, 2000). The formula

(69) |

A pretty formula for is given by

(70) |

sum with sum 1/2 since

(71) |

A particular case of the Wallis formula gives

(72) |

(73) |

coefficient and is the gamma

function (Knopp 1990). Euler obtained

(74) |

follow from for all positive

integers .

An infinite sum due to Ramanujan is

(75) |

*et al.*1989; Borwein and Bailey 2003, p. 109; Bailey

*et al.*

2007, p. 44). Further sums are given in Ramanujan (1913-14),

(76) |

(77) | |||

(78) |

*et al.*1972, Item 139; Borwein

*et al.*1989; Borwein and Bailey 2003, p. 108; Bailey

*et al.*2007, p. 44). Equation (78)

is derived from a modular identity of order 58, although a first derivation was not

presented prior to Borwein and Borwein (1987). The above series both give

(79) |

respectively, about 6 and 8 decimal places per term. Such series exist

because of the rationality of various modular invariants.

The general form of the series is

(80) |

quadratic form discriminant, is the

*j*-function,

(81) | |||

(82) |

series. A class number field involves

th degree algebraic

integers of the constants , , and . Of all series consisting of only integer

terms, the one gives the most numeric digits in the shortest period of time corresponds

to the largest class number 1 discriminant of and was formulated by the Chudnovsky brothers

(1987). The 163 appearing here is the same one appearing in the fact that

(the Ramanujan constant) is very nearly an

integer. Similarly, the factor comes from

the

*j*-function identity for .

The series is given by

(83) | |||

(84) |

*et al.*2007, p. 44). This series gives 14 digits accurately per term. The same equation in another form

was given by the Chudnovsky brothers (1987) and is used by the Wolfram

Language to calculate (Vardi 1991; Wolfram Research),

(85) |

(86) | |||

(87) | |||

(88) |

(89) |

(90) | |||

(91) | |||

(92) |

3 corresponds to and gives 37-38 digits per term.

The fastest converging class number 4 series corresponds

to and is

(93) |

(94) | |||

(95) | |||

(96) |

class number.

A complete listing of Ramanujan's series for found in his

second and third notebooks is given by Berndt (1994, pp. 352-354),

(97) | |||

(98) | |||

(99) | |||

(100) | |||

(101) | |||

(102) | |||

(103) | |||

(104) | |||

(105) | |||

(106) | |||

(107) | |||

(108) | |||

(109) | |||

(110) | |||

(111) | |||

(112) | |||

(113) |

pp. 177-187). Borwein and Borwein (1987b, 1988, 1993) proved other

equations of this type, and

Chudnovsky and Chudnovsky (1987) found similar equations for other

transcendental

constants (Bailey

*et al.*2007, pp. 44-45).

A complete list of independent known equations of this type is given by

(114) | |||

(115) | |||

(116) | |||

(117) | |||

(118) |

(119) | |||

(120) | |||

(121) | |||

(122) |

(123) | |||

(124) |

(125) |

are known (Bailey

*et al.*2007, pp. 45-48).

Bellard gives the exotic formula

(126) |

(127) |

(128) |

hypergeometric function, and transforms it to

(129) |

(130) |

(131) |

(132) |

SEE ALSO: BBP Formula, Digit-Extraction Algorithm, Pi, Pi Approximations,

Pi Continued Fraction, Pi

Digits, Pi Iterations, Pi

Squared, Spigot Algorithm

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CITE THIS AS:

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*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/PiFormulas.html
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