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.
where C is the circumference of a circle, d is the diameter, and r is the radius. 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. More generally,
where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b.
where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and are the arithmetic and geometric iterations of , the arithmetic-geometric mean of a and b with the initial values and .
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.
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 polygons
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:
Integrals
- (integrating two halves to obtain the area of the unit circle)
- [2][note 2] (see also Cauchy distribution)
- (see Dirichlet integral)
- (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 .
Other infinite series
- (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 sum-of-divisors function)
- (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.
- (the original Machin's formula)
Iterative algorithms
- (closely related to Viète's formula)
- (where is the h+1-th entry of m-bit Gray code, )[19]
- (quadratic convergence)[20]
- (cubic convergence)[21]
- (Archimedes' algorithm, see also harmonic mean and geometric mean)[22]
For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.
Hypergeometric inversions
With being the hypergeometric function:
where
and is the sum of two squares function.
Similarly,
where
and is a divisor function.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases.
Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function and the Fourier coefficients of the J-invariant (OEIS: A000521):
where in both cases
Furthermore, by expanding the last expression as a power series in
and setting , we obtain a rapidly convergent series for :[note 3]
Miscellaneous
- (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[23])
- (due to Gauss,[24] is the lemniscate constant)
- (where is the Gauss N-function)
- (where is the principal value of the complex logarithm)[note 4]
- (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 5]
- (where and are Ramanujan's class invariants)[26][note 6]
Notes
The relation was valid until the 2019 revision of the SI.
(integral form of arctan over its entire domain, giving the period of tan)
The coefficients can be obtained by reversing the Puiseux series of
at .
When , this gives algebraic approximations to Gelfond's constant .
When , this gives algebraic approximations to Gelfond's constant .
Other
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.
Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 126
Chrystal, G. (1900). Algebra, an Elementary Text-book: Part II. p. 335.
Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 112
Cooper, Shaun (2017). Ramanujan's Theta Functions (First ed.). Springer. ISBN 978-3-319-56171-4. (page 647)
Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. 1. p. 245
Carl B. Boyer, A History of Mathematics, Chapter 21., pp. 488–489
Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. 1. p. 244
Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. page 49
Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 2
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
Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 248
- Borwein, Peter (2000). "The amazing number π" (PDF). Nieuw Archief voor Wiskunde. 5th series. 1 (3): 254–258. Zbl 1173.01300.
- Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.