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List of formulae involving π
Uses of the constant From Wikipedia, the free encyclopedia
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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
Summarize
Perspective
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:
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Physics
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Perspective
- 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):
- Period of a spring-mass system with spring constant and mass :
- 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.[1] (This gives the digits of π in base b up to N digits past the radix point.)
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Formulae yielding π
Summarize
Perspective
Integrals
- (integrating two halves to obtain the area of the unit circle)
- (integrating a quarter of a circle with a radius of two to obtain )
- [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).
- (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
- (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:[6]
Other infinite series
- (see also Basel problem and Riemann zeta function)
- , where B2n is a Bernoulli number.
- (see Leibniz formula for pi)
In general,
where is the th Euler number.[9]
- (see Gregory coefficients)
- (where is the rising factorial)[10]
- (Nilakantha series)
- (where is the th Fibonacci number)
- (where is the th Lucas number)
- (where is the sum-of-divisors function)
- (where is the number of prime factors of the form of )[13]
The last two formulas are special cases of
which generate infinitely many analogous formulas for when
- (derived from Euler's solution to the Basel problem)
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 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 th Fibonacci number.
whenever and , , are positive real numbers (see List of trigonometric identities). A special case is
Complex functions
The following equivalences are true for any complex :
Also
Suppose a lattice is generated by two periods . We define the quasi-periods of this lattice by and where is the Weierstrass zeta function ( and are in fact independent of ). Then the periods and quasi-periods are related by the Legendre identity:
Continued fractions
- (Ramanujan, is the lemniscate constant)[18]
For more on the fourth identity, see Euler's continued fraction 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.
Asymptotics
- (asymptotic growth rate of the central binomial coefficients)
- (asymptotic growth rate of the Catalan numbers)
- (where is Euler's totient function)
The symbol means that the ratio of the left-hand side and the right-hand side tends to one as .
The symbol means that the difference between the left-hand side and the right-hand side tends to zero as .
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)
- (derived from Euler's solution to Basel problem, see Riemann zeta 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]
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See also
References
Further reading
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