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

:$\pi = \frac Cd$
where is the circumference of a circle, is the diameter. 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 and is the radius. More generally,

:$A = \pi ab$
where is the area enclosed by an ellipse with semi-major axis and semi-minor axis .

:$A=4\pi r^2$
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 and $\operatorname$ is the arithmetic–geometric mean.

:$A=(k+1)(k+2)\pi r^2$
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 redefinition of the SI base units.
::$\mu_0 \approx 4 \pi \cdot 10^\,\mathrm/\mathrm^2$

*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):
::$T=\frac\sqrt$

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

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

A puzzle involving "colliding billiard balls":

Formulae yielding

Integrals

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

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

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

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

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

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

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

:$\int_^\infty \frac x \,dx=\pi$

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

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

:$\int_0^\infty \frac=\frac$ (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)

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

:$\frac \sum^\infty_ \frac=\frac 1 \pi$ (see Chudnovsky algorithm)

:$\frac \sum^\infty_ \frac=\frac 1 \pi$ (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)

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

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.

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

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

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

:$\sum_^\infty \frac = 1 - \frac + \frac - \frac + \frac - \cdots = \arctan = \frac$ (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 $F_n$ is the ''n''-th Fibonacci number)

:$\pi = \sum_^\infty \frac=1 + \frac + \frac + \frac - \frac + \frac + \frac + \frac + \frac - \frac + \frac + \frac - \frac + \cdots$   (where $\epsilon (n)$ 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 $x\in\mathbb\setminus\mathbb.$

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 .

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 exponential formulas

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

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

:$e^z\in\mathbb\leftrightarrow\Im z\in\pi\mathbb$
:$e^z=1\leftrightarrow z\in 2\pi i\mathbb$

Continued fractions

:$\frac = 1 + \cfrac$

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

:$\pi=$

:$\pi = \cfrac$

:$2\pi =$
For more on the fourth identity, see Euler's continued fraction formula.

(See also Continued fraction and Generalized continued fraction.)

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, $h \in \left\$)
:$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)

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

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

Miscellaneous

:$\Gamma (s)\Gamma (1-s)=\frac$ (Euler's reflection formula, see Gamma function)

:$\pi^\Gamma \left(\frac\right)\zeta (s)=\pi^\Gamma\left(\frac\right)\zeta (1-s)$ (the functional equation of the Riemann zeta function)

:$e^=\sqrt$

:$e^=\sqrt$ (where $\zeta (s,a)$ is the Hurwitz zeta function and the derivative is taken with respect to the first variable)

:$\pi =\Beta (1/2,1/2)=\Gamma (1/2)^2$ (see also Beta function)

:$\pi = \frac=\frac$ (where agm is the arithmetic–geometric mean)

:$\pi = \operatorname\left(\theta_2^2(1/e),\theta_3^2(1/e)\right)$ (where $\theta_2$ and $\theta_3$ are the Jacobi theta functions)

:$\pi=-\frac\ln q,\quad k=\frac$ (where $q\in (0,1)$ and $\operatorname(k)$ is the complete elliptic integral of the first kind with modulus $k$; reflecting the nome-modulus inversion problem) page 41

:$\pi =-\frac\ln q,\quad k'=\frac$ (where $q\in (0,1)$)

:$\operatorname(1,\sqrt)=\frac$ (due to Gauss, $\varpi$ is the lemniscate constant)

:$i\pi=\operatorname(-1)=\lim_n\left((-1)^-1\right)$ (where $\operatorname$ is the principal value of the complex logarithm)The $n$th root with the smallest positive principal argument is chosen.

:$1-\frac=\lim_\frac \sum_^n (n\bmod k)$ (where $n\bmod k$ is the remainder upon division of ''n'' by ''k'')

:$\pi = \lim_ \frac \sum_^ \; \sum_^ \begin 1 & \text \sqrt \le r \\ 0 & \text \sqrt > r \end$ (summing a circle's area)

:$\pi = \lim_ \frac \sum_^n \sqrt$ (Riemann sum to evaluate the area of the unit circle)

:$\pi = \lim_\frac=\lim_ \frac = \lim_ \frac\left(\frac\right)^2$ (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 $e^\pi$.

See also

*
*
*
*
*

References

Notes

Other

*.

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, .

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