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mathematical constant π |
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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
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.[1] (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)
- [2][note 2] (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;[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 n-th Fibonacci number)
- (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
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 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)[18]
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, )[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)
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])
- (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
- 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
References
Notes
- ↑ 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 , this gives algebraic approximations to Gelfond's constant .
- ↑ When , this gives algebraic approximations to Gelfond's constant .
Other
- ↑ Galperin, G. (2003). "Playing pool with π (the number π from a billiard point of view)" (PDF). Regular and Chaotic Dynamics. 8 (4): 375–394. doi:10.1070/RD2003v008n04ABEH000252.
- ↑ Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 4
- ↑ A000796 – OEIS
- ↑ 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
- ↑ Gourdon, Xavier. "Computation of the n-th decimal digit of π with low memory" (PDF). Numbers, constants and computation. p. 1.
- ↑ Weisstein, Eric W. "Pi Formulas", MathWorld
- ↑ 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
- ↑ Wästlund, Johan. "Summing inverse squares by euclidean geometry" (PDF). The paper gives the formula with a minus sign instead, but these results are equivalent.
- ↑ 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. - ↑ Rudin, Walter (1987). Real and Complex Analysis (Third ed.). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 3
- 1 2 Loya, Paul (2017). Amazing and Aesthetic Aspects of Analysis. Springer. p. 589. ISBN 978-1-4939-6793-3.
- ↑ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
- ↑ Vellucci, Pierluigi; Bersani, Alberto Maria (2019-12-01). "$$\pi $$-Formulas and Gray code". Ricerche di Matematica. 68 (2): 551–569. arXiv:1606.09597. doi:10.1007/s11587-018-0426-4. ISSN 1827-3491. S2CID 119578297.
- ↑ Abrarov, Sanjar M.; Siddiqui, Rehan; Jagpal, Rajinder K.; Quine, Brendan M. (2021-09-04). "Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π". Mathematics. 9 (17): 2162. arXiv:2107.01027. doi:10.3390/math9172162.
- ↑ 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
- 1 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 41
- ↑ Gilmore, Tomack. "The Arithmetic-Geometric Mean of Gauss" (PDF). Universität Wien. p. 13.
- ↑ Borwein, J.; Borwein, P. (2000). "Ramanujan and Pi". Pi: A Source Book. Springer Link. pp. 588–595. doi:10.1007/978-1-4757-3240-5_62. ISBN 978-1-4757-3242-9.
- ↑ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 248
- Tóth, László (2020), "Transcendental Infinite Products Associated with the +-1 Thue-Morse Sequence" (PDF), Journal of Integer Sequences, 23: 20.8.2, arXiv:2009.02025.
Further reading
- Peter Borwein, The Amazing Number Pi
- Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.