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, 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 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 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]
- 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":
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object.[1] (This gives the digits of π in base b up to N digits past the radix point.)
Formulae yielding π[edit]
Integrals[edit]
- (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 .
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:
- [5]
- (see Bailey–Borwein–Plouffe formula)
Plouffe's series for calculating arbitrary decimal digits of π:[6]
Other infinite series[edit]
- (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]
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where is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.
Machin-like formulae[edit]
- (the original Machin's formula)
Infinite products[edit]
- (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 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 equivalences are true for any complex :
- [16]
Also
Continued fractions[edit]
- [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 algorithms[edit]
- (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.
Asymptotics[edit]
- (asymptotic growth rate of the central binomial coefficients)
- (asymptotic growth rate of the Catalan numbers)
- (Stirling's approximation)
- (where is Euler's totient function)
Hypergeometric inversions[edit]
With being the hypergeometric function, let and
- .
Then
where
Similarly, let and
with being a divisor function. Then
where
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases.
Miscellaneous[edit]
- (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])
- (where and is the complete elliptic integral of the first kind with modulus ; reflecting the nome-modulus inversion problem)[24]
- (where )[24]
- (due to Gauss,[25] 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)[26][note 4]
- (where and are Ramanujan's class invariants)[27][note 5]
See also[edit]
References[edit]
- ^ The relation was valid until the 2019 redefinition of the SI base units.
- ^ (integral form of arctan over its entire domain, giving the period of tan)
- ^ The th root with the smallest positive principal argument is chosen.
- ^ When