List Of Formulae Involving π

The following is a list of significant formulae involving the mathematical constant . Many of these formulae can be found in the article '' Pi'', or the article '' Approximations of ''.

Euclidean geometry

: $\pi = \frac Cd = \frac C$
where is the circumference of a circle, is the diameter, and is the radius. More generally,
: $\pi=\frac$
where and are, respectively, the perimeter and the width of any curve of constant width.

: $A = \pi r^2$
where is the area of a circle. More generally,
: $A = \pi ab$
where is the area enclosed by an ellipse with semi-major axis and semi-minor axis .

: $C=\frac\left(a_1^2-\sum_^\infty 2^(a_n^2-b_n^2)\right)$
where is the circumference of an ellipse with semi-major axis and semi-minor axis and $a_n,b_n$ are the arithmetic and geometric iterations of $\operatorname(a,b)$, the arithmetic-geometric mean of and with the initial values $a_0=a$ and $b_0=b$.

: $A=4\pi r^2$
where is the area between the witch of Agnesi and its asymptotic line; is the radius of the defining circle.

: $A=\frac r^2=\frac$
where is the area of a squircle with minor radius , $\Gamma$ is the gamma function.

: $A=(k+1)(k+2)\pi r^2$
where is the area of an epicycloid with the smaller circle of radius and the larger circle of radius ($k\in\mathbb$), assuming the initial point lies on the larger circle.

: $A=\frac\pi a^2$
where is the area of a rose with angular frequency ($k\in\mathbb$) and amplitude .

: $L=\fracc=\frac$
where is the perimeter of the lemniscate of Bernoulli with focal distance .

: $V = \pi r^3$
where is the volume of a sphere and is the radius.

: $SA = 4\pi r^2$
where is the surface area of a sphere and is the radius.

: $H = \pi^2 r^4$
where is the hypervolume of a 3-sphere and is the radius.

: $SV = 2\pi^2 r^3$
where is the surface volume of a 3-sphere and is the radius.

Regular convex polygons

Sum of internal angles of a regular convex polygon with sides:
: $S=(n-2)\pi$
Area of a regular convex polygon with sides and side length :
: $A=\frac\cot\frac$
Inradius of a regular convex polygon with sides and side length :
: $r=\frac\cot\frac$
Circumradius of a regular convex polygon with sides and side length :
: $R=\frac\csc\frac$

Physics

* The cosmological constant:
*: $\Lambda = \rho$

* Heisenberg's uncertainty principle:
*: $\Delta x\, \Delta p \ge \frac h$

* Einstein's field equation of general relativity:
*: $R_ - \fracg_R + \Lambda g_ = T_$

* Coulomb's law for the electric force in vacuum:
*: $F = \frac$

* Magnetic permeability of free space:The relation $\mu_0 = 4 \pi \cdot 10^\,\mathrm/\mathrm^2$ was valid until the 2019 revision of the SI.
*: $\mu_0 \approx 4 \pi \cdot 10^\,\mathrm/\mathrm^2$

* Approximate period of a simple pendulum with small amplitude:
*: $T \approx 2\pi \sqrt\frac L g$

* Exact period of a simple pendulum with amplitude $\theta_0$ ($\operatorname$ is the arithmetic–geometric mean):
*: $T=\frac\sqrt$

* Period of a spring-mass system with spring constant $k$ and mass $m$:
*: $T = 2\pi \sqrt$

* Kepler's third law of planetary motion:
*: $\frac = \frac$

* The buckling formula:
*: $F =\frac$

A puzzle involving "colliding billiard balls":
: $\lfloor\rfloor$
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 ''b''^2''N''''m'', 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

: $2 \int_^1 \sqrt\,dx = \pi$ (integrating two halves $y(x)=\sqrt$ to obtain the area of the unit circle)

: $\int_^2 \sqrt\,dx = \pi$ (integrating a quarter of a circle with a radius of two $x^2+y^2=4$ to obtain $/4$)

: $\int_^\infty \operatornamex \, dx = \pi$

: $\int_^\infty \int_t^\infty e^ \, dx \, dt = \int_^\infty \int_t^\infty e^ \, dx \, dt = \pi$

: $\int_^1\frac = \pi$

: $\int_^\infty\frac = \pi$(integral form of arctan over its entire domain, giving the period of tan) (see also Cauchy distribution)

: $\int_^\infty \frac \,dx = \pi$ (see Dirichlet integral)

: $\int_^\infty e^\,dx = \sqrt$ (see Gaussian integral).

: $\oint\frac z = 2\pi i$ (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).

: $\int_0^\infty \ln\left(1+\frac\right)\, dx=\pi$

: $\int_0^1 \,dx = - \pi$ (see also Proof that 22/7 exceeds ).

: $\int_0^1 \,dx = \pi-$

: $\int_0^\infty \frac\, dx=\frac,\quad 0<\alpha<1$

: $\int_0^\infty \frac=\frac$ (where $\operatorname$ is the arithmetic–geometric mean; see also elliptic integral)

Note that with symmetric integrands $f(-x)=f(x)$, formulas of the form $\int_^af(x)\,dx$ can also be translated to formulas $2\int_^af(x)\,dx$.

Efficient infinite series

: $\sum_^\infty \frac = \sum_^\infty\frac = \frac \pi 2$ (see also Double factorial)

: $\sum_^\infty \frac=\frac$

: $\sum_^\infty \frac=\frac$

: $\sum^\infty_ \frac=\frac$ (see Chudnovsky algorithm)

: $\sum^\infty_ \frac=\frac$ (see Srinivasa Ramanujan, Ramanujan–Sato series)

The following are efficient for calculating arbitrary binary digits of :
: $\sum_^\infty \frac\left(\frac+\frac+\frac\right)=\pi$

: $\sum_^ \frac \left( \frac - \frac - \frac - \frac\right)=\pi$ (see Bailey–Borwein–Plouffe formula)

: $\sum_^\infty \frac\left(\frac+\frac+\frac-\frac\right)=2\pi$

: $\sum_^ \frac \left( - \frac - \frac + \frac - \frac - \frac - \frac + \frac \right)=2^6\pi$

Plouffe's series for calculating arbitrary decimal digits of :

: $\sum_^\infty k\frac=\pi +3$

Other infinite series

: $\zeta(2) = \frac + \frac + \frac + \frac + \cdots = \frac$ (see also Basel problem and Riemann zeta function)

: $\zeta(4)= \frac + \frac + \frac + \frac + \cdots = \frac$

: $\zeta(2n) = \sum_^ \frac\, = \frac + \frac + \frac + \frac + \cdots = (-1)^\frac$ , where ''B''_2''n'' is a Bernoulli number.

: $\sum_^\infty \frac\, \zeta(n+1) = \pi$

: $\sum_^\infty \frac\, \zeta(n+1) = (1+\sqrt)\pi$

: $\sum_^\infty \frac(\zeta (n)-1)=\ln \pi$

: $\sum_^\infty \zeta (2n)\frac=\ln\frac,\quad 0<, x, <1$

: $\sum_^\infty \frac = 1 - \frac + \frac - \frac + \frac - \cdots = \arctan = \frac$ (see Leibniz formula for pi)

: $\sum_^\infty \frac=1+\frac13-\frac15-\frac17+\frac19+\frac-\cdots=\frac$ ( Newton, ''Second Letter to Oldenburg'', 1676)

: $\sum_^\infty \frac=1-\frac+\frac-\frac+\frac-\cdots =\sqrt\arctan\frac=\frac$ ( Madhava series)

: $\sum_^\infty \frac=\frac - \frac + \frac - \frac + \cdots=\frac$

: $\sum_^\infty \frac1 = \frac + \frac + \frac + \frac + \cdots = \frac$

: $\sum_^\infty \left( \frac \right)^2 = \frac + \frac + \frac + \frac + \cdots = \frac$

: $\sum_^\infty \left( \frac \right)^3 = \frac - \frac + \frac - \frac + \cdots = \frac$

: $\sum_^\infty \left( \frac \right)^4 = \frac + \frac + \frac + \frac + \cdots = \frac$

: $\sum_^\infty \left( \frac \right)^5 = \frac - \frac + \frac - \frac + \cdots = \frac$

: $\sum_^\infty \left( \frac \right)^6 = \frac + \frac + \frac + \frac + \cdots = \frac$

In general,
: $\sum_^\infty \frac=(-1)^\frac\left(\frac\right)^,\quad k\in\mathbb_0$
where $E_$ is the $2k$th Euler number.

: $\sum_^\infty \binom\frac = 1 - \frac - \frac-\cdots = \frac$

: $\sum_^\infty \frac = \frac+\frac +\frac +\cdots=\frac$

: $\sum_^\infty (-1)^\left, G_\=, G_1, +, G_2, -, G_4, -, G_5, +, G_7, +, G_8, -, G_, -, G_, +\cdots =\frac$ (see Gregory coefficients)

: $\sum_^\infty \frac\sum_^\infty \frac=\frac$ (where $(x)_n$ is the rising factorial)

: $\sum_^\infty \frac=\pi -3$ ( Nilakantha series)

: $\sum_^\infty \frac=\frac$ (where $F_2n$ is the $2n$th Fibonacci number)

: $\sum_^\infty \frac=\frac$ (where $L_n$ is the $n$th Lucas number)

: $\sum_^\infty \sigma(n)e^=\frac-\frac$ (where $\sigma$ is the sum-of-divisors function)

: $\pi = \sum_^\infty \frac=1 + \frac + \frac + \frac - \frac + \frac + \frac + \frac + \frac - \frac + \frac + \frac - \frac + \cdots$   (where $\varepsilon (n)$ is the number of prime factors of the form $p\equiv 1\,(\mathrm\,4)$ of $n$)

: $\frac=\sum_^\infty \frac=1+\frac-\frac+\frac+\frac-\frac-\frac+\frac+\frac+\cdots$   (where $\varepsilon (n)$ is the number of prime factors of the form $p\equiv 3\, (\mathrm\, 4)$ of $n$)

: $\pi=\sum_^\infty \frac$

: $\pi^2=\sum_^\infty \frac$

The last two formulas are special cases of
: $\begin\frac&=\sum_^\infty \frac\\ \left(\frac\right)^2&=\sum_^\infty \frac\end$
which generate infinitely many analogous formulas for $\pi$ when $x\in\mathbb\setminus\mathbb.$

: $\pi=\sqrt$ (derived from Euler's solution to the Basel problem)

Some formulas relating and harmonic numbers are given here. Further infinite series involving π are:

where $(x)_n$ is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Machin-like formulae

: $\frac = \arctan 1$

: $\frac = \arctan\frac + \arctan\frac$

: $\frac = 2 \arctan\frac - \arctan\frac$

: $\frac = 2 \arctan\frac + \arctan\frac$

: $\frac = 4 \arctan\frac - \arctan\frac$ (the original Machin's formula)

: $\frac = 5 \arctan\frac + 2 \arctan\frac$

: $\frac = 6 \arctan\frac + 2 \arctan\frac + \arctan\frac$

: $\frac = 12 \arctan\frac + 32 \arctan\frac - 5 \arctan\frac + 12 \arctan\frac$

: $\frac = 44 \arctan\frac + 7 \arctan\frac - 12 \arctan\frac + 24 \arctan\frac$

Infinite products

: $\frac = \left(\prod_\frac\right)\cdot\left( \prod_\frac\right)=\frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots,$ (Euler)
where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.

: $\frac=\left(\displaystyle\prod_ \frac\right) \cdot \left(\displaystyle\prod_ \frac\right)=\frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots$

: $\frac=\prod_^ \frac = \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots$ (see also Wallis product)

: $\frac=\prod_^\infty\left(1+\frac\right)^=\left(1+\frac\right)^\left(1+\frac\right)^\left(1+\frac\right)^\cdots$ (another form of Wallis product)

Viète's formula:
: $\frac=\frac2 \cdot \frac2 \cdot \frac2 \cdot \cdots$

A double infinite product formula involving the Thue–Morse sequence:
: $\frac=\prod_ \prod_ \left( \frac \right) ^,$
where $\epsilon_n = (-1)^$ and $t_n$ is the Thue–Morse sequence .

Infinite product representation :
: $\frac= \prod_^ \frac$

Arctangent formulas

: $\frac=\arctan \frac, \qquad\qquad k\geq 2$
: $\frac=\sum_\arctan \frac,$
where $a_k=\sqrt$ such that $a_1=\sqrt$.

: $\frac = \sum_^\infty \arctan\frac = \arctan\frac + \arctan\frac + \arctan\frac + \arctan\frac + \cdots$

where $F_k$ is the $k$th Fibonacci number.

: $\pi =\arctan a+\arctan b+\arctan c$
whenever $a+b+c=abc$ and $a$, $b$, $c$ are positive real numbers (see List of trigonometric identities). A special case is
: $\pi =\arctan 1+\arctan 2+\arctan 3.$

Complex functions

: $e^ +1 = 0$ (Euler's identity)

The following equivalences are true for any complex $z$:

: $e^z\in\mathbb\leftrightarrow\Im z\in\pi\mathbb$
: $e^z=1\leftrightarrow z\in 2\pi i\mathbb$
Also
: $\frac=\lim_\sum_^N \frac-\frac,\quad z\in\mathbb.$

Suppose a lattice $\Omega$ is generated by two ''periods'' $\omega_1,\omega_2$. We define the ''quasi-periods'' of this lattice by $\eta_1=\zeta (z+\omega_1;\Omega)-\zeta (z;\Omega)$ and $\eta_2=\zeta (z+\omega_2;\Omega)-\zeta (z;\Omega)$ where $\zeta$ is the Weierstrass zeta function ($\eta_1$ and $\eta_2$ are in fact independent of $z$). Then the periods and quasi-periods are related by the ''Legendre identity'':
: $\eta_1\omega_2-\eta_2\omega_1=2\pi i.$

Continued fractions

: $\frac = 1 + \cfrac$

: $\frac= \quad$ ( Ramanujan, $\varpi$ is the lemniscate constant)

: $\pi=$

: $\pi = \cfrac$

: $2\pi =$

: $\pi=4-\cfrac$
For more on the fourth identity, see Euler's continued fraction formula.

Iterative algorithms

: $a_0=1,\, a_=\left(1+\frac\right)a_n,\, \pi=\lim_\frac$

: $a_1=0,\, a_=\sqrt,\, \pi =\lim_ 2^n\sqrt$ (closely related to Viète's formula)
: $\omega(i_n,i_,\dots,i_)=2+i_ \sqrt=\omega(b_n,b_,\dots,b_),\, i_ \in\, \, b_k=\begin 0& \text i_k=1\\ 1& \text i_k=-1 \end, \, \pi=$ (where $g_$ is the h+1-th entry of m-bit Gray code, $h \in \left\$)

: $\forall k \in \mathbb, \, a_1 = 2^, \, a_ = a_n + 2^(1 - \tan (2^ a_n)), \, \pi = 2^ \lim _ a_n$ (quadratic convergence)

: $a_1=1,\, a_=a_n+\sin a_n,\, \pi =\lim_a_n$ (cubic convergence)

: $a_0=2\sqrt,\, b_0=3,\, a_=\operatorname(a_n,b_n),\, b_=\operatorname(a_,b_n),\, \pi =\lim_a_n=\lim_b_n$ ( Archimedes' algorithm, see also harmonic mean and geometric mean)

For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.

Asymptotics

: $\binom\sim \frac$ (asymptotic growth rate of the central binomial coefficients)

: $C_n\sim \frac$ (asymptotic growth rate of the Catalan numbers)

: $n! \sim \sqrt \left(\frac\right)^n$ (Stirling's approximation)

: $\log n!\simeq \left(n+\frac12\right)\log n-n+\frac$

: $\sum_^ \varphi (k) \sim \frac$ (where $\varphi$ is Euler's totient function)

: $\sum_^ \frac \sim \frac$

The symbol $\sim$ means that the ''ratio'' of the left-hand side and the right-hand side tends to one as $n\to\infty$.

The symbol $\simeq$ means that the ''difference'' between the left-hand side and the right-hand side tends to zero as $n\to\infty$.

Hypergeometric inversions

With $_2F_1$ being the hypergeometric function:
: $\sum_^\infty r_2(n)q^n=_2F_1\left(\frac12,\frac12,1,z\right)$
where
: $q=\exp\left(-\pi \frac = \lim_ \frac\left(\frac\right)^2$ (by combining Stirling's approximation with Wallis product)

: $\pi=\lim_\frac\ln\frac$ (where $\lambda$ is the modular lambda function)When $n\in\mathbb^+$, this gives algebraic approximations to Gelfond's constant $e^\pi$.

: $\pi=\lim_\frac\ln \left(2^ G_n\right)=\lim_\frac\ln \left(2^g_n\right)$ (where $G_n$ and $g_n$ are Ramanujan's class invariants) p. 248When $\sqrt\in\mathbb^+$, this gives algebraic approximations to Gelfond's constant $e^\pi$.
: $\pi=\lim_\frac \left(\frac\right)^4$ where $\mu_1^'= \frac$ and $\mu_2^'= \frac$ are the first and second raw moments o
Pavan's
Probability Mass Function $p(x_i) =\begin \frac \quad \text &x_i = \binom, \binom,\binom,...,\binom,&i=0,1,2,..., \\ \frac \quad \text& x_i = \binom, \quad i = \frac \end \text$

See also

*
*
*
*
*

References

Notes

Other

* .

Further reading

*
* Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: ''Number Theory 1: Fermat's Dream''. American Mathematical Society, Providence 1993, .

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