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.

 

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

 

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.

PhysicsEdit

 
 
 
 
 
  • Period of a simple pendulum with small amplitude:
 
 
 

Formulae yielding πEdit

IntegralsEdit

  (integrating two halves   to obtain the area of a circle of radius  )
 
 
 
  (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).

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:

 [2]
  (see Bailey–Borwein–Plouffe formula)
 

Plouffe's series for calculating arbitrary decimal digits of π:[3]

 

Other infinite seriesEdit

    (see also Basel problem and Riemann zeta function)
 
  , where B2n is a Bernoulli number.
 [4]
    (see Leibniz formula for pi)
  (Madhava series)
 
 
 
 
 
 
 
 
 
  (see Gregory coefficients)
  (where   is the rising factorial)[5]
  (Nilakantha series)
  (where   is the n-th Fibonacci number)
    (where   is the number of prime factors of the form   of  ; Euler, 1748)[6]

Some formulas relating π and harmonic numbers are given here.

Machin-like formulaeEdit

 
 
 
 
  (the original Machin's formula)
 
 
 
 
 

where   is the n-th Fibonacci number.

Infinite seriesEdit

Some infinite series involving π are:[7]

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

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

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)

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  .

 

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

 

Continued fractionsEdit

 
 
 
 

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

(See also Continued fraction and Generalized continued fraction.)

MiscellaneousEdit

  (Stirling's approximation)
  (Euler's identity)
  (see Euler's totient function)
  (see 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[8])
  (where   and   is the complete elliptic integral of the first kind with modulus  ; reflecting the nome-modulus inversion problem)[9]
  (where  )[9]
  (due to Gauss,[10]   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 Stirling's approximation)
  (recurrence form of the above formula)
  (closely related to Viète's formula)
  (cubic convergence)[11]
  (Archimedes' algorithm, see also harmonic mean and geometric mean)

See alsoEdit

ReferencesEdit

  1. ^ https://oeis.org/A000796
  2. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 126
  3. ^ Gourdon, Xavier. "Computation of the n-th decimal digit of π with low memory" (PDF). Numbers, constants and computation. p. 1.
  4. ^ Weisstein, Eric W. "Pi Formulas", MathWorld
  5. ^ Cooper, Shaun (2017). Ramanujan’s Theta Functions (First ed.). Springer. ISBN 978-3-319-56171-4. (page 647)
  6. ^ Carl B. Boyer, A History of Mathematics, Chapter 21., pp. 488–489
  7. ^ 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.
  8. ^ 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
  9. ^ 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
  10. ^ Gilmore, Tomack. "The Arithmetic-Geometric Mean of Gauss" (PDF). Universität Wien. p. 13.
  11. ^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 49

Further readingEdit