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Pi

Table of contents: Definitions - Numerical value - Euler's identity - Elementary function representations - Integral representations - Series representations - Product representations - Limit representations - Special function representations - Approximations

Definitions

b5d706
Symbol: Pi — π\pi — The constant pi (3.14...)

Numerical value

6505a9
π∈[3.1415926535897932384626433832795028841971693993751±5.83⋅10−51]\pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right]
47acde
Table of simple expressions involving π\pi to 50 digits
0c838a
π∉Q\pi \notin \mathbb{Q}
155575
π∉Q‾\pi \notin \overline{\mathbb{Q}}
483547
π=∑n=1∞A000796 ⁣(n)101−n\pi = \sum_{n=1}^{\infty} \text{A000796}\!\left(n\right) {10}^{1 - n}

Euler's identity

271314
eπi+1=0{e}^{\pi i} + 1 = 0

Elementary function representations

0c9939
π=4atan⁡(1)\pi = 4 \operatorname{atan}(1)
3ff35f
π=2acos⁡(0)\pi = 2 \operatorname{acos}(0)
722241
π=2asin⁡(1)\pi = 2 \operatorname{asin}(1)
b89166
π=zero*⁡ x∈[3,4]sin⁡(x)\pi = \mathop{\operatorname{zero*}\,}\limits_{x \in \left[3, 4\right]} \sin(x)
590136
π=−ilog⁡(−1)\pi = -i \log(-1)
030560
π=10asin⁡ ⁣(12φ)\pi = 10 \operatorname{asin}\!\left(\frac{1}{2 \varphi}\right)

Machin-type formulas

f8d280
π=16atan⁡ ⁣(15)−4atan⁡ ⁣(1239)\pi = 16 \operatorname{atan}\!\left(\frac{1}{5}\right) - 4 \operatorname{atan}\!\left(\frac{1}{239}\right)
cbf396
π=4atan⁡ ⁣(12)+4atan⁡ ⁣(13)\pi = 4 \operatorname{atan}\!\left(\frac{1}{2}\right) + 4 \operatorname{atan}\!\left(\frac{1}{3}\right)
b1357b
π=8atan⁡ ⁣(12)−4atan⁡ ⁣(17)\pi = 8 \operatorname{atan}\!\left(\frac{1}{2}\right) - 4 \operatorname{atan}\!\left(\frac{1}{7}\right)
0644b6
π=8atan⁡ ⁣(13)+4atan⁡ ⁣(17)\pi = 8 \operatorname{atan}\!\left(\frac{1}{3}\right) + 4 \operatorname{atan}\!\left(\frac{1}{7}\right)
5278da
π=4atan⁡ ⁣(12)+4atan⁡ ⁣(15)+4atan⁡ ⁣(18)\pi = 4 \operatorname{atan}\!\left(\frac{1}{2}\right) + 4 \operatorname{atan}\!\left(\frac{1}{5}\right) + 4 \operatorname{atan}\!\left(\frac{1}{8}\right)
7ce79e
π=4atan⁡ ⁣(13)+4atan⁡ ⁣(14)+4atan⁡ ⁣(17)+4atan⁡ ⁣(113)\pi = 4 \operatorname{atan}\!\left(\frac{1}{3}\right) + 4 \operatorname{atan}\!\left(\frac{1}{4}\right) + 4 \operatorname{atan}\!\left(\frac{1}{7}\right) + 4 \operatorname{atan}\!\left(\frac{1}{13}\right)
8332d8
π=48atan⁡ ⁣(149)+128atan⁡ ⁣(157)−20atan⁡ ⁣(1239)+48atan⁡ ⁣(1110443)\pi = 48 \operatorname{atan}\!\left(\frac{1}{49}\right) + 128 \operatorname{atan}\!\left(\frac{1}{57}\right) - 20 \operatorname{atan}\!\left(\frac{1}{239}\right) + 48 \operatorname{atan}\!\left(\frac{1}{110443}\right)

Integral representations

464961
π=2∫−111−x2 dx\pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx
fc8149
π=∫−1111−x2 dx\pi = \int_{-1}^{1} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx
04cd99
π=∫−∞∞1x2+1 dx\pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx
dae4a7
π=(∫−∞∞e−x2 dx)2\pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2}
81f500
π=227−∫01x4(1−x)41+x2 dx\pi = \frac{22}{7} - \int_{0}^{1} \frac{{x}^{4} {\left(1 - x\right)}^{4}}{1 + {x}^{2}} \, dx
bd3faa
π=355113−13164∫01x8(1−x)8(25+816x2)1+x2 dx\pi = \frac{355}{113} - \frac{1}{3164} \int_{0}^{1} \frac{{x}^{8} {\left(1 - x\right)}^{8} \left(25 + 816 {x}^{2}\right)}{1 + {x}^{2}} \, dx
9a3503
π=∫−∞∞sinc⁡(x) dx\pi = \int_{-\infty}^{\infty} \operatorname{sinc}(x) \, dx
8107d6
π=∫−∞∞sinc⁡2 ⁣(x) dx\pi = \int_{-\infty}^{\infty} \operatorname{sinc}^{2}\!\left(x\right) \, dx
5033c7
π=2e∫0∞cos⁡(x)x2+1 dx\pi = 2 e \int_{0}^{\infty} \frac{\cos(x)}{{x}^{2} + 1} \, dx
6ed553
π=8(∫0∞sin⁡ ⁣(x2) dx)2\pi = 8 {\left(\int_{0}^{\infty} \sin\!\left({x}^{2}\right) \, dx\right)}^{2}
859856
π=8(∫0∞cos⁡ ⁣(x2) dx)2\pi = 8 {\left(\int_{0}^{\infty} \cos\!\left({x}^{2}\right) \, dx\right)}^{2}
d8cb3e
π=∫0∞θ2 ⁣(0,it) dt\pi = \int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \, dt
e00d9e
π=3∫0∞(θ3 ⁣(0,it)−1) dt\pi = 3 \int_{0}^{\infty} \left(\theta_{3}\!\left(0 , i t\right) - 1\right) \, dt

Series representations

f617c0
π=4∑n=0∞(−1)n2n+1\pi = 4 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{2 n + 1}
93831d
π=∑n=0∞2n+1(n!)2(2n+1)!\pi = \sum_{n=0}^{\infty} \frac{{2}^{n + 1} {\left(n !\right)}^{2}}{\left(2 n + 1\right)!}
419b45
π=∑n=0∞n!(2n+1)!!\pi = \sum_{n=0}^{\infty} \frac{n !}{\left(2 n + 1\right)!!}
fddfe6
π=∑n=0∞116n(48n+1−28n+4−18n+5−18n+6)\pi = \sum_{n=0}^{\infty} \frac{1}{{16}^{n}} \left(\frac{4}{8 n + 1} - \frac{2}{8 n + 4} - \frac{1}{8 n + 5} - \frac{1}{8 n + 6}\right)
6b9f81
1π=229801∑n=0∞(4n)!(1103+26390n)(n!)4⋅3964n\frac{1}{\pi} = \frac{2 \sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{\left(4 n\right)! \left(1103 + 26390 n\right)}{{\left(n !\right)}^{4} \cdot {396}^{4 n}}
57fcaf
1π=12∑n=0∞(−1)n(6n)!(13591409+545140134n)(3n)!(n!)3⋅6403203n+3/2\frac{1}{\pi} = 12 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}
0479f5
π=72∑n=1∞1n(eπn−1)−96∑n=1∞1n(e2πn−1)+24∑n=1∞1n(e4πn−1)\pi = 72 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{\pi n} - 1\right)} - 96 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{2 \pi n} - 1\right)} + 24 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{4 \pi n} - 1\right)}
338055
π=8∑n=0∞1(4n+1)(4n+3)\pi = 8 \sum_{n=0}^{\infty} \frac{1}{\left(4 n + 1\right) \left(4 n + 3\right)}
fbc53d
π26=∑n=1∞1n2\frac{{\pi}^{2}}{6} = \sum_{n=1}^{\infty} \frac{1}{{n}^{2}}
11302a
π212=∑n=1∞(−1)n+1n2\frac{{\pi}^{2}}{12} = \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n + 1}}{{n}^{2}}
9bf21b
π490=∑n=1∞1n4\frac{{\pi}^{4}}{90} = \sum_{n=1}^{\infty} \frac{1}{{n}^{4}}
8dff72
π=2∑n=0∞atan⁡ ⁣(1n2+n+1)\pi = 2 \sum_{n=0}^{\infty} \operatorname{atan}\!\left(\frac{1}{{n}^{2} + n + 1}\right)
31eecc
π=2∑n=0∞atan⁡ ⁣(1F2n+1)\pi = 2 \sum_{n=0}^{\infty} \operatorname{atan}\!\left(\frac{1}{F_{2 n + 1}}\right)
bad5d9
π=3(3∑n=0∞(−1)n3n+1−log⁡(2))\pi = \sqrt{3} \left(3 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{3 n + 1} - \log(2)\right)
54c80d
π=42∑n=0∞(−1)n4n+1−2log⁡ ⁣(1+2)\pi = 4 \sqrt{2} \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{4 n + 1} - 2 \log\!\left(1 + \sqrt{2}\right)
f78fa0
π=3(92∑n=0∞1(2nn)−6)\pi = \sqrt{3} \left(\frac{9}{2} \sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} - 6\right)
dbdf08
π=3(92∑n=1∞n(2nn)−3)\pi = \sqrt{3} \left(\frac{9}{2} \sum_{n=1}^{\infty} \frac{n}{{2 n \choose n}} - 3\right)
a2e6f9
π=∑n=1∞(3n−1)ζ ⁣(n+1)4n\pi = \sum_{n=1}^{\infty} \frac{\left({3}^{n} - 1\right) \zeta\!\left(n + 1\right)}{{4}^{n}}

Product representations

69fe63
π=2∏n=1∞4n24n2−1\pi = 2 \prod_{n=1}^{\infty} \frac{4 {n}^{2}}{4 {n}^{2} - 1}
490cf4
π=2∏n=2∞sec⁡ ⁣(π2n)\pi = 2 \prod_{n=2}^{\infty} \sec\!\left(\frac{\pi}{{2}^{n}}\right)
a91200
π26=∏p(1−1p2)−1\frac{{\pi}^{2}}{6} = \prod_{p} {\left(1 - \frac{1}{{p}^{2}}\right)}^{-1}
6fce07
2π=∏n=1∞an2   where a1=2,  an=2+an−1\frac{2}{\pi} = \prod_{n=1}^{\infty} \frac{a_{n}}{2}\; \text{ where } a_{1} = \sqrt{2},\;a_{n} = \sqrt{2 + a_{n - 1}}

Limit representations

dea83d
π=lim⁡n→∞4n2∑k=0nn2−k2\pi = \lim_{n \to \infty} \frac{4}{{n}^{2}} \sum_{k=0}^{n} \sqrt{{n}^{2} - {k}^{2}}
e1e106
π=lim⁡n→∞16nn(2nn)2\pi = \lim_{n \to \infty} \frac{{16}^{n}}{n {{2 n \choose n}}^{2}}
420007
π=lim⁡n→∞12((−1)n+1(2n)!B2n)1/(2n)\pi = \lim_{n \to \infty} \frac{1}{2} {\left({\left(-1\right)}^{n + 1} \frac{\left(2 n\right)!}{B_{2 n}}\right)}^{1 / \left(2 n\right)}
220e8d
3π2=lim⁡N→∞1N2∑n=1Nφ(n)\frac{3}{{\pi}^{2}} = \lim_{N \to \infty} \frac{1}{{N}^{2}} \sum_{n=1}^{N} \varphi(n)
6d9ceb
π=4(agm⁡ ⁣(1,12))21−∑j=0∞2jcj2=lim⁡n→∞(an+bn)21−∑j=0n2jcj2   where (an,bn)=agm⁡n ⁣(1,12),  cn=an−bn\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

Special function representations

8fab22
π=(Γ ⁣(12))2\pi = {\left(\Gamma\!\left(\frac{1}{2}\right)\right)}^{2}
2371b9
π=32Γ ⁣(13)Γ ⁣(23)\pi = \frac{\sqrt{3}}{2} \Gamma\!\left(\frac{1}{3}\right) \Gamma\!\left(\frac{2}{3}\right)
63ba30
π=12Γ ⁣(14)Γ ⁣(34)\pi = \frac{1}{\sqrt{2}} \Gamma\!\left(\frac{1}{4}\right) \Gamma\!\left(\frac{3}{4}\right)
67bb53
π=6ζ ⁣(2)\pi = \sqrt{6 \zeta\!\left(2\right)}
591d64
π=B ⁣(12,12)\pi = \mathrm{B}\!\left(\frac{1}{2}, \frac{1}{2}\right)
033c51
π=G2 ⁣(i)\pi = G_{2}\!\left(i\right)
dabb47
π=12(Γ ⁣(14))4/3(agm⁡ ⁣(1,2))2/3\pi = \frac{1}{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4 / 3} {\left(\operatorname{agm}\!\left(1, \sqrt{2}\right)\right)}^{2 / 3}
ce5423
π=2K(0)\pi = 2 K(0)
07e35f
π=2E(0)\pi = 2 E(0)
9206a3
π=6Li⁡2 ⁣(1)\pi = \sqrt{6 \operatorname{Li}_{2}\!\left(1\right)}
1448e3
π=2 2F1 ⁣(12,12,32,1)\pi = 2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, 1\right)
a7095f
1π=12 2F1 ⁣(12,−12,1,1)\frac{1}{\pi} = \frac{1}{2} \,{}_2F_1\!\left(\frac{1}{2}, -\frac{1}{2}, 1, 1\right)
c6c108
1π=14 2F1 ⁣(−12,−12,1,1)\frac{1}{\pi} = \frac{1}{4} \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, 1, 1\right)
2a0316
π=2 2F1 ⁣(−12,−12,12,1)\pi = 2 \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, 1\right)
f55b36
π=4( 2F1 ⁣(−12,1,12,−1)−1)\pi = 4 \left(\,{}_2F_1\!\left(-\frac{1}{2}, 1, \frac{1}{2}, -1\right) - 1\right)
769f6e
π=2 2F1 ⁣(1,1,12,12)−4\pi = 2 \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{1}{2}\right) - 4
488a30
π=4(2 2F1 ⁣(−12,−12,12,12)−1)\pi = 4 \left(\sqrt{2} \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) - 1\right)
826257
π=3(92 2F1 ⁣(1,1,12,14)−6)\pi = \sqrt{3} \left(\frac{9}{2} \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{1}{4}\right) - 6\right)
3d276b
π=12 2F1 ⁣(−12,−12,12,14)−63\pi = 12 \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{1}{4}\right) - 6 \sqrt{3}
2806fd
π=923 2F1 ⁣(1,1,32,14)\pi = \frac{9}{2 \sqrt{3}} \,{}_2F_1\!\left(1, 1, \frac{3}{2}, \frac{1}{4}\right)
68b73d
1π=239 2F1 ⁣(−13,13,1,1)\frac{1}{\pi} = \frac{2 \sqrt{3}}{9} \,{}_2F_1\!\left(-\frac{1}{3}, \frac{1}{3}, 1, 1\right)
42d727
π=5φ+22φ 2F1 ⁣(1,1,32,1(2φ)2)\pi = \frac{5 \sqrt{\varphi + 2}}{2 \varphi} \,{}_2F_1\!\left(1, 1, \frac{3}{2}, \frac{1}{{\left(2 \varphi\right)}^{2}}\right)
8ee7c9
π=ψ′ ⁣(14)−8G\pi = \sqrt{\psi'\!\left(\frac{1}{4}\right) - 8 G}
f56273
π=4L ⁣(1,χ4 . 3)\pi = 4 L\!\left(1, \chi_{4 \, . \, 3}\right)

Approximations

2516c2
∣π−227∣<0.00127\left|\pi - \frac{22}{7}\right| < 0.00127
1e3a25
∣π−355113∣<2.67⋅10−7\left|\pi - \frac{355}{113}\right| < 2.67 \cdot 10^{-7}
fdc3a3
∣π−log⁡ ⁣(6403203+744)163∣<2.24⋅10−31\left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| < 2.24 \cdot 10^{-31}
4c0698
∣1π−(12∑n=0N−1(−1)n(6n)!(13591409+545140134n)(3n)!(n!)3⋅6403203n+3/2)∣<1151931373056000N\left|\frac{1}{\pi} - \left(12 \sum_{n=0}^{N - 1} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}
13c539
∣π−(an+bn)21−∑j=0n2jcj2∣≤2n+8e−π2n+1   where (an,bn)=agm⁡n ⁣(1,12),  cn=an−bn\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC