List of formulae involving π


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 geometryEdit

 

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

 

where L 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 A is the area between the witch of Agnesi and its asymptotic line; r is the radius of the defining circle.

 

where A is the area of a squircle with minor radius r,   is the gamma function and   is the arithmetic–geometric mean.

 

where A is the area of an epicycloid with the smaller circle of radius r and the larger circle of radius kr ( ), assuming the initial point lies on the larger circle.

 

where A is the area of a rose with angular frequency k ( ) and amplitude a.

 

where L is the perimeter of the lemniscate of Bernoulli with focal distance c.

 

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 polygonsEdit

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:

 

PhysicsEdit

 
 
 
 
 
  • Approximate period of a simple pendulum with small amplitude:
 
  • Exact period of a simple pendulum with amplitude   (  is the arithmetic–geometric mean):
 
 
 

A puzzle involving "colliding billiard balls":[1]

Formulae yielding πEdit

IntegralsEdit

  (integrating two halves   to obtain the area of the unit circle)
 
 
 
 [2][note 2] (see also Cauchy distribution)
  (see Gaussian integral).
  (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).
 [3]
 
  (see also Proof that 22/7 exceeds π).
 
  (where   is the arithmetic–geometric mean;[4] see also elliptic integral)

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

Efficient infinite seriesEdit

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

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

 [5]
  (see Bailey–Borwein–Plouffe formula)
 

Plouffe's series for calculating arbitrary decimal digits of π:[6]

 

Other infinite seriesEdit

  (see also Basel problem and Riemann zeta function)
 
  , where B2n is a Bernoulli number.
 [7]
 
 
  (see Leibniz formula for pi)
  (Newton, Second Letter to Oldenburg, 1676)[8]
  (Madhava series)
 
 
 
 
 
 
 

In general,

 

where   is the  th Euler number.[9]

 
 
  (see Gregory coefficients)
  (where   is the rising factorial)[10]
  (Nilakantha series)
  (where   is the n-th Fibonacci number)
    (where   is the number of prime factors of the form   of  )[11][12]
    (where   is the number of prime factors of the form   of  )[13]
 
 [14]

The last two formulas are special cases of

 

which generate infinitely many analogous formulas for   when  

Some formulas relating π and harmonic numbers are given here. Further infinite series involving π are:[15]

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

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

Machin-like formulaeEdit

 
 
 
 
  (the original Machin's formula)
 
 
 
 

Infinite productsEdit

  (Euler)
where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
 
  (see also Wallis product)
  (another form of 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 formulasEdit

 
 

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 formulasEdit

  (Euler's identity)

The following equivalences are true for any complex  :

 
 [16]

Continued fractionsEdit

 [17]
  (Ramanujan,   is the lemniscate constant)[18]
 [17]
 
 

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

(See also Continued fraction and Generalized continued fraction.)

Iterative algorithmsEdit

 
  (closely related to Viète's formula)
  (where   is the h+1-th entry of m-bit Gray code,  )[19]
  (cubic convergence)[20]
  (Archimedes' algorithm, see also harmonic mean and geometric mean)[21]

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

AsymptoticsEdit

  (asymptotic growth rate of the central binomial coefficients)
  (asymptotic growth rate of the Catalan numbers)
  (Stirling's approximation)
  (where   is Euler's totient function)
 

MiscellaneousEdit

  (Euler's reflection formula, see Gamma function)
  (the functional equation of the Riemann zeta function)
 
  (where   is the Hurwitz zeta function and the derivative is taken with respect to the first variable)
  (see also Beta function)
  (where agm is the arithmetic–geometric mean)
  (where   and   are the Jacobi theta functions[22])
  (where   and   is the complete elliptic integral of the first kind with modulus  ; reflecting the nome-modulus inversion problem)[23]
  (where  )[23]
  (due to Gauss,[24]   is the lemniscate constant)
  (where   is the principal value of the complex logarithm)[note 3]
  (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)[25][note 4]
  (where   and   are Ramanujan's class invariants)[26][note 5]

See alsoEdit

ReferencesEdit

NotesEdit

  1. ^ The relation   was valid until the 2019 redefinition of the SI base units.
  2. ^ (integral form of arctan over its entire domain, giving the period of tan)
  3. ^ The  th root with the smallest positive principal argument is chosen.
  4. ^ When  , this gives algebraic approximations to Gelfond's constant  .
  5. ^ When  , this gives algebraic approximations to Gelfond's constant  .

OtherEdit

  1. ^ Galperin, G. (2003). "Playing pool with π (the number π from a billiard point of view)" (PDF). Regular and Chaotic Dynamics. 8 (4): 375–394. doi:10.1070/RD2003v008n04ABEH000252.
  2. ^ Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 4
  3. ^ A000796 – OEIS
  4. ^ 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
  5. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 126
  6. ^ Gourdon, Xavier. "Computation of the n-th decimal digit of π with low memory" (PDF). Numbers, constants and computation. p. 1.
  7. ^ Weisstein, Eric W. "Pi Formulas", MathWorld
  8. ^ Chrystal, G. (1900). Algebra, an Elementary Text-book: Part II. p. 335.
  9. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 112
  10. ^ Cooper, Shaun (2017). Ramanujan's Theta Functions (First ed.). Springer. ISBN 978-3-319-56171-4. (page 647)
  11. ^ Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. 1. p. 245
  12. ^ Carl B. Boyer, A History of Mathematics, Chapter 21., pp. 488–489
  13. ^ Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. 1. p. 244
  14. ^ Wästlund, Johan. "Summing inverse squares by euclidean geometry" (PDF). The paper gives the formula with a minus sign instead, but these results are equivalent.
  15. ^ 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.
  16. ^ Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 3
  17. ^ a b Loya, Paul (2017). Amazing and Aesthetic Aspects of Analysis. Springer. p. 589. ISBN 978-1-4939-6793-3.
  18. ^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
  19. ^ Vellucci, Pierluigi; Bersani, Alberto Maria (2019-12-01). "$$\pi $$-Formulas and Gray code". Ricerche di Matematica. 68 (2): 551–569. arXiv:1606.09597. doi:10.1007/s11587-018-0426-4. ISSN 1827-3491. S2CID 119578297.
  20. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 49
  21. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 2
  22. ^ 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
  23. ^ 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
  24. ^ Gilmore, Tomack. "The Arithmetic-Geometric Mean of Gauss" (PDF). Universität Wien. p. 13.
  25. ^ Borwein, J.; Borwein, P. (2000). Ramanujan and Pi. Springer Link. pp. 588–595. doi:10.1007/978-1-4757-3240-5_62. ISBN 978-1-4757-3242-9.
  26. ^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 248

Further readingEdit