Pi Formulas
000
$$\pi = \pi$$
001
$$\pi = 180^{\circ}$$
002
$$\pi = 4 \operatorname{atan}(1)$$
003
$$\pi = 16 \operatorname{acot}(5) - 4 \operatorname{acot}(239)$$
004
$$\pi = 48 \operatorname{acot}(49) + 128 \operatorname{acot}(57)
- 20 \operatorname{acot}(239) + 48 \operatorname{acot}(110443)$$
005
$$\pi = 2i \log\left(\frac{1-i}{1+i}\right)$$
006
$$\pi = -2i \operatorname{asinh}(i)$$
007
$$\pi = \operatorname{Ci}(-\infty) / i$$
$\operatorname{Ci}(z)$ is the cosine integral function.
008
$$\pi = \Gamma(1/2)^2$$
009
$$\pi = B(1/2,1/2)$$
$B(a,b)$ is the beta function.
010
$$\pi = (2 / \operatorname{erf}'(0))^2$$
$\operatorname{erf}(z)$ is the error function.
011
$$\pi = c, \quad \sin(c) = 0, c \in [3,4]$$
012
$$\pi = 2c, \quad \cos(c) = 0, c \in [1,2]$$
013
$$\pi = -2iW(-\pi/2)$$
$W(z)$ is the Lambert W-function.
014
$$\pi = j_{1/2,1}, \quad J_{\nu}(j_{\nu,n}) = 0$$
$J_{\nu}(z)$ is a Bessel function.
015
$$\pi = \frac{3 \sqrt{3}}{2\,{}_2F_{1}(-1/3,1/3,1,1)}$$
016
$$\pi = 8 \left(\frac{\Gamma(5/4)}{\Gamma(3/4) {}_2F_1(1/2,1/2,1,1/2)}\right)^2$$
017
$$\pi = 4 \left(\frac{{}_1F_2(1,3/2,1,1)}{\mathbf{L}_{-1/2}(2)} \right)^2$$
$\mathbf{L}_{\nu}(z)$ is a Struve function.
018
$$\pi = \frac{1}{{{G_{0, 2}^{1, 0}\left(\begin{matrix} & \\0 & \frac{1}{2} \end{matrix} \middle| {0} \right)}}^{2}}$$
$G$ is the Meijer G-function.
027
$$\pi = \sqrt{6} \sqrt{\zeta\left(2\right)}$$
028
$$\pi = \sqrt{6 \zeta\left(2, 3\right) + \frac{15}{2}}$$
029
$$\pi = \sqrt{8 C + \zeta(2, 3/4)}$$
$C$ is Catalan's constant.
031
$$\pi = 2 \sqrt{3} \sqrt{\eta\left(2\right)}$$
$\eta(s)$ is the Dirichlet eta function.
036
$$\pi = \sqrt{12} \sqrt{- \operatorname{Li}_{2}\left(-1\right)}$$
037
$$\pi = \sqrt{6 \log^{2}{\left (2 \right )} + 12 \operatorname{Li}_{2}\left(\frac{1}{2}\right)}$$
042
$$\pi = \sum_{k=0}^{\infty} \frac{4 \left(-1\right)^{k}}{2 k + 1}$$
046
$$\pi = \frac{3 \sqrt{3}}{2} \sum_{k=0}^{\infty} \frac{k!^{2}}{\left(2 k + 1\right)!}$$
059
$$\pi = \int_{-\infty}^{\infty} \frac{1}{x^{2} + 1}\, dx$$
060
$$\pi = \int_{-\infty}^{\infty} e^{- x^{2}}\, dx^{2}$$
061
$$\pi = 2 \int_{-1}^{1} \sqrt{- x^{2} + 1}\, dx$$
063
$$\pi = - \frac{3}{2} \int_{-1}^{1}\int_{-1}^{1} \frac{1}{\sqrt{x^{2} + y^{2} + 1}}\, dx\, dy + 6 \log{\left (\sqrt{3} + 2 \right )}$$
064
$$\pi = 2 \sqrt{2} \sqrt{\int_{0}^{1}\int_{0}^{1} \frac{1}{1 - x^{2} y^{2}}\, dx\, dy}$$
065
$$\pi = \sqrt{6} \sqrt{\int_{0}^{1}\int_{0}^{1} \frac{1}{1 - x y}\, dx\, dy}$$
066
$$\pi = \sqrt{6} \sqrt{\int_{0}^{\infty} \frac{x}{e^{x} - 1}\, dx}$$
067
$$\pi = \int_{0}^{1} \frac{16 x - 16}{x^{4} - 2 x^{3} + 4 x - 4}\, dx$$
068
$$\pi = \frac{3 \sqrt{3}}{4} + 24 \int_{0}^{\frac{1}{4}} \sqrt{x - x^{2}}\, dx$$
069
$$\pi = - \int_{0}^{1} \frac{x^{4} \left(x - 1\right)^{4}}{x^{2} + 1}\, dx + \frac{22}{7}$$
070
$$\pi = - \frac{1}{3164} \int_{0}^{1} \frac{x^{8} \left(x - 1\right)^{8}}{x^{2} + 1} \left(816 x^{2} + 25\right)\, dx + \frac{355}{113}$$
071
$$\pi = \int_{0}^{\infty} \frac{2 \sin(x)}{x}\, dx$$
072
$$\pi = \frac{40}{11} \int_{0}^{\infty} \frac{\sin^{6}{\left (x \right )}}{x^{6}} \, dx$$
073
$$\pi = e \int_{-\infty}^{\infty} \frac{\cos{\left (x \right )}}{x^{2} + 1}\, dx$$
074
$$\pi = 8 \int_{0}^{\infty} \cos{\left (x^{2} \right )}\, dx^{2}$$
076
$$\pi = \frac{1}{2} e^{2 \int_{0}^{1} \log \Gamma{\left (x \right )}\, dx}$$
077
$$\pi = 2 \prod_{k=2}^{\infty} \sec{\left (2^{- k} \pi \right )}$$
080
$$\pi = \prod_{k=1}^{\infty} \frac{4 k^{2}}{\left(2 k - 1\right) \left(2 k + 1\right)}$$
081
$$\pi = 2 \prod_{k=1}^{\infty} \frac{4 k^{2}}{4 k^{2} - 1}$$
$$\pi = \sqrt{6} \sqrt{\log{\left (\prod_{k=1}^{\infty} e^{\frac{1}{k^{2}}} \right )}}$$
083
$$\pi = \frac{\prod_{k=2}^{\infty} \frac{k^{2} - 1}{k^{2} + 1}}{\operatorname{csch}{\left (\pi \right )}}$$
084
$$\pi = \sinh{\left (\pi \right )} \prod_{k=2}^{\infty} \frac{k^{2} - 1}{k^{2} + 1}$$
085
$$\pi = \frac{1}{\sinh{\left (\pi \right )}} \left(\cosh{\left (\sqrt{2} \pi \right )} - \cos{\left (\sqrt{2} \pi \right )}\right) \prod_{k=2}^{\infty} \frac{k^{4} - 1}{k^{4} + 1}$$
086
$$\pi = \frac{\sinh{\left (\pi \right )}}{4 \prod_{k=2}^{\infty} \left(1 - \frac{1}{k^{4}}\right)}$$
087
$$\pi = \frac{\sinh{\left (\pi \right )}}{2 \prod_{k=2}^{\infty} \left(1 + \frac{1}{k^{2}}\right)}$$
088
$$\pi = \frac{e^{\gamma + 2}}{2 \left(\prod_{k=1}^{\infty} \left(1 + \frac{1}{k}\right)^k e^{\frac{1}{2 k} - 1}\right)^{2}}$$
089
$$\pi = \frac{3 \sqrt{2} \cosh^{2}{\left (\frac{\sqrt{3} \pi}{2} \right )} \operatorname{csch}{\left (\sqrt{2} \pi \right )}}{\prod_{k=1}^{\infty} \frac{\left(1 + \frac{1}{k} + \frac{1}{k^{2}}\right)^{2}}{1 + \frac{2}{k} + \frac{3}{k^{2}}}}$$
090
$$\pi = \frac{2}{e} \prod_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^{\left(-1\right)^{k + 1} k}$$
091
$$\pi = \lim_{k \to \infty}\left(\frac{16^{k}}{k {\binom{2 k}{k}}^{2}}\right)$$
092
$$\pi = \lim_{x \to \infty}\left(2 \operatorname{Si}{\left (2 x \right )}\right)$$
093
$$\pi = \log{\left (4 \right )} / \log{\left (\lim_{n \to \infty} \prod_{k=n}^{2 n} \frac{\pi}{2 \operatorname{atan}{\left (k \right )}} \right )}$$
094
$$\pi = \lim_{k \to \infty}\left(\frac{2^{4 k + 1} k!^{4}}{\left(2 k + 1\right) \left(2 k\right)!^{2}}\right)$$
095
$$\pi = \frac{1}{2} \lim_{k \to \infty}\left(\frac{\left(\frac{k}{e}\right)^{- k}}{\sqrt{k}} k!\right)^{2}$$
096
$$\pi = \lim_{k \to \infty} \left(- \frac{\left(-1\right)^{k}}{\left(2 k\right)!} 2^{2 k - 1} B_{2 k}\right)^{- \frac{1}{2 k}}$$
098
$$\pi = \lim_{x \to \infty}\left(2 \sqrt[4]{x} e^{\frac{2 x^{\frac{3}{2}}}{3}} \operatorname{Ai}\left(x\right)\right)^{-2}$$
099
$$\pi = \lim_{x \to \infty}\left(\sqrt[4]{x} e^{- \frac{2 x^{\frac{3}{2}}}{3}} \operatorname{Bi}\left(x\right)\right)^{-2}$$
100
$$\pi = \sqrt{6} \sqrt{\sum_{k=1}^{\infty} \frac{1}{k^{2}}}$$
101
$$\pi = \operatorname{acos}{\left (-1 \right )}$$
102
$$\pi = 4 \sum_{k=1}^{\infty} \operatorname{atan}\left(\frac{1}{F_{2k+1}}\right)$$
103
$$\pi = \int_{0}^{\infty} \frac{3 x^{2} + 6}{\left(x^{2} + 1\right) \left(x^{2} + 4\right)}\, dx$$
104
$$\pi = - i \log{\left (-1 \right )}$$
105
$$\pi = \int_{-1}^{1} \frac{1}{\sqrt{- x^{2} + 1}}\, dx$$
106
$$\pi = \operatorname{arg}(-1)$$
107
$$\pi = \frac{\Gamma(1/4) \Gamma(3/4)}{\sqrt{2}}$$
108
$$\pi = \int_{0}^{1} \frac{3 \sqrt{3}}{2(1 - x^3)^\frac{1}{3}} dx$$
109
$$\pi = \left( \int_{0}^{\infty} 2 e^{-x^2 + 1} \cos(2x) dx \right) ^2 $$
110
$$\pi = \int_{-\infty}^{\infty} \frac{\sqrt{2}}{1 + x^4}$$
111
$$\pi = \int_{0}^{\infty} \frac{x^2}{(x^2 + (\frac{1}{4})^2)^2} dx$$