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 geometry
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
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
- Coulomb's law for the electric force in vacuum:
- Approximate period of a simple pendulum with small amplitude:
- Exact period of a simple pendulum with amplitude ( is the arithmetic–geometric mean):
- The buckling formula:
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. (This gives the digits of π in base b up to N digits past the radix point.)
Formulae yielding π
Integrals
- (integrating two halves to obtain the area of the unit circle)
- (see also Cauchy distribution)
- (see Gaussian integral).
- (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).
- (see also Proof that 22/7 exceeds π).
- (where is the arithmetic–geometric mean; see also elliptic integral)
Note that with symmetric integrands , formulas of the form can also be translated to formulas .
Efficient infinite series
- (see also Double factorial)
- (see Chudnovsky algorithm)
The following are efficient for calculating arbitrary binary digits of π:
Plouffe's series for calculating arbitrary decimal digits of π:
Other infinite series
- (see also Basel problem and Riemann zeta function)
- , where B2n is a Bernoulli number.
- (see Leibniz formula for pi)
- (Newton, Second Letter to Oldenburg, 1676)
In general,
where is the th Euler number.
- (see Gregory coefficients)
- (where is the rising factorial)
- (Nilakantha series)
- (where is the n-th Fibonacci number)
- (where is the number of prime factors of the form of )
- (where is the number of prime factors of the form of )
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:
where is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.
Machin-like formulae
- (the original Machin's formula)
Infinite products
- (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)
A double infinite product formula involving the Thue–Morse sequence:
- where and is the Thue–Morse sequence (Tóth 2020).
Arctangent formulas
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
The following equivalences are true for any complex :
Also
Continued fractions
- (Ramanujan, is the lemniscate constant)
For more on the fourth identity, see Euler's continued fraction formula.
(See also Continued fraction and Generalized continued fraction.)
Iterative algorithms
- (closely related to Viète's formula)
- (where is the h+1-th entry of m-bit Gray code, )
- (quadratic convergence)
- (cubic convergence)
- (Archimedes' algorithm, see also harmonic mean and geometric mean)
For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.
Asymptotics
- (asymptotic growth rate of the central binomial coefficients)
- (asymptotic growth rate of the Catalan numbers)
- (where is Euler's totient function)
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)
- (where and is the complete elliptic integral of the first kind with modulus ; reflecting the nome-modulus inversion problem)
- (where )
- (due to Gauss, is the lemniscate constant)
- (where is the principal value of the complex logarithm)
- (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)
- (where and are Ramanujan's class invariants)
See also
- List of mathematical identities
- Lists of mathematics topics
- List of trigonometric identities – Equalities that involve trigonometric functions
- List of topics related to π – Topics related to the mathematical constant
- List of representations of e