List of formulae involving π

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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.

Euclidean geometry[edit]

where C is the circumference of a circle, d is the diameter. More generally,

where s and w are, respectively, the perimeter and the width of any curve of constant width.

where A is the area of a circle and r is the radius. More generally,

where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b.

where V is the volume of a sphere and r is the radius.

where SA is the surface area of a sphere and r is the radius.

where H is the hypervolume of a 3-sphere and r is the radius.

where SV is the surface volume of a 3-sphere and r is the radius.

Regular convex polygons[edit]

Sum S of internal angles of a regular convex polygon with n sides:

Area A of a regular convex polygon with n sides and side length s:

Inradius r of a regular convex polygon with n sides and side length s:

Circumradius R of a regular convex polygon with n sides and side length s:

Physics[edit]

  • Period of a simple pendulum with small amplitude:

Formulae yielding π[edit]

Integrals[edit]

(integrating two halves to obtain the area of the unit circle)
(integral form of arctan over its entire domain, giving the period of tan).
(see Gaussian integral).
(when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).
[1]
(see also Proof that 22/7 exceeds π).
(where is the arithmetic–geometric mean;[2] see also elliptic integral)

Note that with symmetric integrands , formulas of the form can also be translated to formulas .

Efficient infinite series[edit]

(see also Double factorial)
(see Chudnovsky algorithm)
(see Srinivasa Ramanujan, Ramanujan–Sato series)

The following are efficient for calculating arbitrary binary digits of π:

[3]
(see Bailey–Borwein–Plouffe formula)

Plouffe's series for calculating arbitrary decimal digits of π:[4]

Other infinite series[edit]

  (see also Basel problem and Riemann zeta function)
, where B2n is a Bernoulli number.
[5]
  (see Leibniz formula for pi)
(Madhava series)
(see Gregory coefficients)
(where is the rising factorial)[6]
(Nilakantha series)
(where is the n-th Fibonacci number)
  (where is the number of prime factors of the form of ; Euler, 1748)[7]

Some formulas relating π and harmonic numbers are given here.

Machin-like formulae[edit]

(the original Machin's formula)

Infinite series[edit]

Some infinite series involving π are:[8]

where is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Infinite products[edit]

(Euler)
where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
[citation needed]
(see also Wallis product)

Viète's formula:

A double infinite product formula involving the Thue–Morse sequence:

where and is the Thue–Morse sequence (Tóth 2020).

Arctangent formulas[edit]

where such that .

where is the k-th Fibonacci number.

whenever and , , are positive real numbers (see List of trigonometric identities). A special case is

Complex exponential formulas[edit]

(Euler's identity)

The following implications are true for any complex :

Continued fractions[edit]

For more on the third identity, see Euler's continued fraction formula.

(See also Continued fraction and Generalized continued fraction.)

Iterative algorithms[edit]

(closely related to Viète's formula)
(where is the h+1-th entry of m-bit Gray code, )[9]
(cubic convergence)[10]
(Archimedes' algorithm, see also harmonic mean and geometric mean)[11]

For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.

Miscellaneous[edit]

(where is the principal value of the complex logarithm)
( is the Hurwitz zeta function and the derivative is taken with respect to the first variable)
(asymptotic growth rate of the central binomial coefficients)
(asymptotic growth rate of the Catalan numbers)
(Stirling's approximation)
(where is Euler's totient function)
(see also Beta function and Gamma function)
(where agm is the arithmetic–geometric mean)
(where and are the Jacobi theta functions[12])
(where  and is the complete elliptic integral of the first kind with modulus ; reflecting the nome-modulus inversion problem)[13]
(where )[13]
(due to Gauss,[14] is the lemniscate constant)
(where is the remainder upon division of n by k)
(summing a circle's area)
(Riemann sum to evaluate the area of the unit circle)
(by combining Stirling's approximation with Wallis product)
(where is the modular lambda function)[15][note 1]
(where and are Ramanujan's class invariants)[16][note 2]

See also[edit]

References[edit]

Notes[edit]

  1. ^ When , this gives algebraic approximations to Gelfond's constant .
  2. ^ When , this gives algebraic approximations to Gelfond's constant .

Other[edit]

  1. ^ A000796 - OEIS
  2. ^ Carson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
  3. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 126
  4. ^ Gourdon, Xavier. "Computation of the n-th decimal digit of π with low memory" (PDF). Numbers, constants and computation. p. 1.
  5. ^ Weisstein, Eric W. "Pi Formulas", MathWorld
  6. ^ Cooper, Shaun (2017). Ramanujan's Theta Functions (First ed.). Springer. ISBN 978-3-319-56171-4. (page 647)
  7. ^ Carl B. Boyer, A History of Mathematics, Chapter 21., pp. 488–489
  8. ^ Simon Plouffe / David Bailey. "The world of Pi". Pi314.net. Retrieved 2011-01-29.
    "Collection of series for π". Numbers.computation.free.fr. Retrieved 2011-01-29.
  9. ^ Vellucci, Pierluigi; Bersani, Alberto Maria (2019-12-01). "$$\pi $$-Formulas and Gray code". Ricerche di Matematica. 68 (2): 551–569. doi:10.1007/s11587-018-0426-4. ISSN 1827-3491.
  10. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 49
  11. ^ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 2
  12. ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. page 225
  13. ^ a b Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. page 41
  14. ^ Gilmore, Tomack. "The Arithmetic-Geometric Mean of Gauss" (PDF). Universität Wien. p. 13.
  15. ^ Borwein, J.; Borwein, P. "Ramanujan and Pi". Springer Link.
  16. ^ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435. p. 240

Further reading[edit]