Mathematical constant

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A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π (spoken as "pi") means the ratio of the length of a circle's circumference to its diameter. This value is always the same for any circle. In contrast to physical constants, mathematical constants do not come from physical measurements.

Constants and series[change | edit source]

--Tables structure--

  • Value numerical of the constante.
  • LaTeX: Formula or series in TeX format.
  • Formula: For use in programs like Mathematica or Wolfram Alpha.
  • OEIS: Link to: On-Line Encyclopedia of Integer Sequences (OEIS), where the constants are available with more details.
  • Continued fraction: In the simple form [to integer; frac1, frac2, frac3, ...] (in brackets if periodic)
  • Tipo:

You can choose the order of the list by clicking on the name, value, OEIS, etc..

Value Name Symbol LaTeX Formula Type OEIS Continued fraction
3.24697960371746706105000976800847962 Silver, Tutte–Beraha constant  \varsigma  2+2 \cos(2\pi/7)= \textstyle 2+\frac{2+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}{1+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}} 2+2 cos(2Pi/7) T A116425 [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]
1.09864196439415648573466891734359621 Paris constant  C_{Pa}  \prod_{n=2}^\infty \frac{2 \varphi}{\varphi+ \varphi_n}\; , \varphi= {Fi} I A105415 [1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...]
2.74723827493230433305746518613420282 Ramanujan nested radical R5  R_{5} \scriptstyle \sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}}\;=\textstyle\frac{2+\sqrt{5}+\sqrt{15-6\sqrt{5}}}{2} (2+sqrt(5)+sqrt(15-6 sqrt(5)))/2 I [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
2.23606797749978969640917366873127624 Square root of 5, Gauss sum  \sqrt{5}  \scriptstyle \forall \, n=5, \displaystyle  \sum_{k=0}^{n-1} e^{\frac{2 k^2 \pi i}{n}} = 1 + e^\frac{2 \pi i} {5} + e^\frac{8 \pi i} {5} + e^\frac{18 \pi i} {5} + e^\frac{32 \pi i} {5} Sum[k=0 to 4]{e^(2k^2 pi i/5)} I A002163 [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]
= [2;(4),...]
3.62560990822190831193068515586767200 Gamma(1/4) \Gamma(\tfrac14)  4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)! 4(1/4)! T A068466 [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]
0.18785964246206712024851793405427323 MRB constant, Marvin Ray Burns  C_{_{MRB}}  \sum_{n=1}^{\infty} ({-}1)^n (n^{1/n}{-}1) =  - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \sqrt[4]{4}\,... Sum[n=1 to ∞]{(-1)^n (n^(1/n)-1)} T A037077 [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]
0.11494204485329620070104015746959874 Kepler–Bouwkamp constant {\rho}  \prod_{n=3}^\infty \cos\left(\frac\pi n\right) = \cos\left(\frac\pi 3\right) \cos\left(\frac\pi 4\right) \cos\left(\frac\pi 5\right) ... prod[n=3 to ∞]{cos(pi/n)} T A085365 [0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...]
1.78107241799019798523650410310717954 Exp(gamma)
G-Barnes function
e^{\gamma} \prod_{n=1}^\infty \frac{e^{\frac{1}{n}}}{1+\tfrac1n} = \prod_{n=0}^\infty \left(\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac{1}{n+1}} =

\textstyle \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/3} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/4}
\left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/5}...

Prod[n=1 to ∞]{e^(1/n)}/{1 + 1/n} T A073004 [1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...]
1.28242712910062263687534256886979172 Glaisher–Kinkelin constant {A}  e^{\frac{1}{12}-\zeta^{\prime}(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^{\infty} \frac{1}{n+1} \sum\limits_{k=0}^{n} \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)} e^(1/2-zeta´{-1}) T A074962 [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...]
7.38905609893065022723042746057500781 Schwarzschild conic constant e^2  \sum_{n = 0}^\infty \frac{2^n}{n!} = 1+2+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!}+\frac{2^5}{5!}+... Sum[n=0 to ∞]{2^n/n!} T A072334 [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]
= [7,2,(1,1,n,4*n+6,n+2)], n = 3, 6, 9, etc.
1.01494160640965362502120255427452028 Gieseking constant {G_{Gi}} \frac{3\sqrt{3}}{4} \left(1- \sum_{n=0}^\infty \frac{1}{(3n+2)^2}+ \sum_{n=1}^\infty\frac{1}{(3n+1)^2} \right)=

\textstyle \frac{3\sqrt{3}}{4} \left( 1 - \frac{1}{2^2} + \frac{1}{4^2}-\frac{1}{5^2}+\frac{1}{7^2}-\frac{1}{8^2}+\frac{1}{10^2} \pm ... \right).

T A143298 [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]
2.62205755429211981046483958989111941 Lemniscata constant {\varpi}  \pi \, {G} = 4 \sqrt{\tfrac2\pi} \,(\tfrac14 !)^2 4 sqrt(2/pi) (1/4!)^2 T A062539 [2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]
0.83462684167407318628142973279904680 Gauss constant {G}   \underset{ Agm:\; Arithmetic-geometric \; mean} {\frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}} (4 sqrt(2)(1/4!)^2)/pi^(3/2) T A014549 [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]
1.01734306198444913971451792979092052 Zeta(6) \zeta(6) \frac{\pi^6}{945} = \prod_{n=1}^\infty \underset{p_{n}: \, {primo}}\frac{1}{{1-p_n}^{-6}} = \frac{1}{1{-}2^{-6}}{\cdot}\frac{1}{1{-}3^{-6}}{\cdot}\frac{1}{1{-}5^{-6}} ... Prod[n=1 to ∞] {1/(1-ithprime(n)^-6)} T A013664 [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]
0,60792710185402662866327677925836583 Constante de Hafner-Sarnak-McCurley \frac{1}{\zeta(2)}  \frac{6}{\pi^2} {=} \prod_{n = 0}^\infty \underset{p_{n}: \, {primo}}{\left(1- \frac{1}{{p_n}^2}\right)}{=}\textstyle  \left(1{-}\frac{1}{2^2}\right)\left(1{-}\frac{1}{3^2}\right)\left(1{-}\frac{1}{5^2}\right)... Prod{n=1 to ∞} (1-1/ithprime(n)^2) T A059956 [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]
1.11072073453959156175397024751517342 The ratio of a square and circumscribed or inscribed circles \frac{\pi}{2\sqrt 2} \sum_{n = 1}^\infty \frac{(-1)^{\lfloor \frac{n-1}{2}\rfloor}}{2n+1} = \frac{1}{1} + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - ... sum[n=1 to ∞]{(-1)^(floor((n-1)/2))/(2n-1)} T A093954 [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...]
2.80777024202851936522150118655777293 Fransén–Robinson constant {F} \int_{0}^\infty \frac{1}{\Gamma(x)}\, dx. = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, dx N[int[0 to ∞] {1/Gamma(x)}] T A058655 [2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...]
1.64872127070012814684865078781416357 Square root of the number e \sqrt e \sum_{n = 0}^\infty \frac{1}{2^n n!} = \sum_{n = 0}^\infty \frac{1}{(2n)!!} = \frac{1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{48}+\cdots sum[n=0 to ∞]{1/(2^n n!)} T A019774 [1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...]
= [1;1,(1,1,4p+1)], p∈ℕ
i Imaginary number {i} \sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1 sqrt(-1) C
262537412640768743.999999999999250073 Hermite-Ramanujan constant {R}  e^{\pi\sqrt{163}} e^(π sqrt(163)) T A060295 [262537412640768743;1,1333462407511,1,8,1,1,5,...]
4.81047738096535165547303566670383313 John constant  \gamma \sqrt[i]{i} = i^{-i} = i^{\frac{1}{i}} = (i^i)^{-1} = e^{\frac{\pi}{2}} e^(π/2) T A042972 [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,...]
4.53236014182719380962768294571666681 Constante de Van der Pauw  \alpha \frac{\pi}{ln(2)} = \frac{\sum_{n = 0}^\infty \frac{4(-1)^n}{2n+1}} {\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}} = \frac{\frac{4}{1} {-} \frac{4}{3} {+} \frac{4}{5} {-} \frac{4}{7} {+} \frac{4}{9} - ...} {\frac{1}{1}{-}\frac{1}{2}{+}\frac{1}{3}{-}\frac{1}{4}{+}\frac{1}{5}-...} π/ln(2) T A163973 [4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...]
0.76159415595576488811945828260479359 Hyperbolic tangent (1) th \, 1 \frac{e-\frac{1}{e}}{e+\frac{1}{e}} = \frac{e^2-1}{e^2+1} (e-1/e)/(e+1/e) T A073744 [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...]
= [0;(2p+1)], p∈ℕ
0.69777465796400798200679059255175260 Continued Fraction constant {C}_{CF}  \underset{J_{k}() {Bessel}}\underset{{Function}}\frac{J_1(2)}{J_0(2)} = \frac{ \sum\limits_{n = 0}^{\infty} \frac{n}{n!n!}} {{ \sum\limits_{n = 0}^{\infty} \frac{1}{n!n!}}} = \frac{\frac{0}{1}+\frac{1}{1}+\frac{2}{4}+\frac{3}{36}+\frac{4}{576}+...} {\frac{1}{1}+\frac{1}{1}+\frac{1}{4}+\frac{1}{36}+\frac{1}{576}+...} (sum {n=0 to inf} n/(n!n!)) /(sum {n=0 to inf} 1/(n!n!)) A052119 [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]
= [0;(p+1)], p∈ℕ
0.36787944117144232159552377016146086 Inverse Napier constant \frac{1}{e} \sum_{n = 0}^\infty \frac{(-1)^n}{n!} = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + ... sum[n=2 to ∞]{(-1)^n/n!} T A068985 [0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...]
= [0;2,1,(1,2p,1)], p∈ℕ
2.71828182845904523536028747135266250 Napier constant e \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots Sum[n=0 to ∞]{1/n!} T A001113 [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...]
= [2;(1,2p,1)], p∈ℕ
0.49801566811835604271369111746219809
- 0.15494982830181068512495513048388 i
Factorial of i i\,!  \Gamma (1+i) = i \, \Gamma (i) Gamma(1+i) C A212877
A212878
[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...]
- [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i
0.43828293672703211162697516355126482
+ 0.36059247187138548595294052690600 i
Infinite
Tetration of i
 {}^\infty i  \lim_{n \to \infty}  {}^n i  =  \lim_{n \to \infty}  \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n  i^i^i^... C A077589
A077590
[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...]
+ [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i
0.56755516330695782538461314419245334 Module of
Infinite
Tetration of i
|{}^\infty i |  \lim_{n \to \infty} \left | {}^n i \right |  =\left | \lim_{n \to \infty}  \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n  \right | Mod(i^i^i^...) A212479 [0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]
0.26149721284764278375542683860869585 Meissel-Mertens constant M \lim_{n \rightarrow \infty } \left( 
\sum_{p \leq n} \frac{1}{p}  - \ln(\ln(n)) \right) ..... p: primes A077761 [0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...]
1.9287800... Wright constant \omega \left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \right \rfloor = primos: \quad \left\lfloor 2^\omega\right\rfloor =3, \left\lfloor 2^{2^\omega} \right\rfloor =13, \left\lfloor 2^{2^{2^\omega}} \right\rfloor =16381, \dots A086238 [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
0.37395581361920228805472805434641641 Artin constant C_{Artin} \prod_{n=1}^{\infty} \left(1-\frac{1}{p_n(p_n-1)}\right) ...... pn: primo T A005596 [0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...]
4.66920160910299067185320382046620161 Feigenbaum constant δ {\delta}  \lim_{n \to \infty}\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1}} \qquad \scriptstyle x \in (3,8284;\, 3,8495)

 \scriptstyle  x_{n+1}=\,ax_n(1-x_n)\quad {o} \quad x_{n+1}=\,a\sin(x_n)

T A006890 [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]
2.50290787509589282228390287321821578 Feigenbaum constant α \alpha \lim_{n \to \infty}\frac {d_n}{d_{n+1}} T A006891 [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]
5.97798681217834912266905331933922774 Hexagonal Madelung Constant 2 H_{2}(2)  \pi \ln(3) \sqrt 3 Pi Log[3]Sqrt[3] T A086055 [5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...]
0.96894614625936938048363484584691860 Beta(3) \beta (3)  \frac{\pi^3}{32} = \sum_{n=1}^\infty\frac{-1^{n+1}}{(-1+2n)^3} = \frac{1}{1^3} {-} \frac{1}{3^3} {+} \frac{1}{5^3} {-} \frac{1}{7^3} {+} ... Sum[n=1 to ∞]{(-1)^(n+1)/(-1+2n)^3} T A153071 [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]
1.902160583104 Brun constant 2 = Σ inverse twin primes B_{\,2}  \textstyle \sum \underset{p,\, p+2: \, {primos}}{(\frac1{p}+\frac1{p+2})} = (\frac1{3} {+} \frac1{5}) + (\tfrac1{5} {+} \tfrac1{7}) + (\tfrac1{11} {+} \tfrac1{13}) + ... A065421 [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]
0.870588379975 Brun constant 4 = Σ inverse of twin prime B_{\,4} \underset{p,\, p+2,\, p+4,\, p+6: \, {primes}}  {\left(\tfrac1{5} + \tfrac1{7} + \tfrac1{11} + \tfrac1{13}\right)}+ \left(\tfrac1{11} + \tfrac1{13} + \tfrac1{17} + \tfrac1{19}\right)+ \dots A213007 [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]
22.4591577183610454734271522045437350 pi^e \pi^{e} \pi^{e} pi^e A059850 [22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...]
3.14159265358979323846264338327950288 Pi, Archimedes constant \pi \lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\text{...} +\sqrt{2}}}}}_n Sum[n=0 to ∞]{(-1)^n 4/(2n+1)} T A000796 [3;7,15,1,292,1,1,1,2,1,3,1,14,...]
0.06598803584531253707679018759684642 e^{-e} e^{-e} ... Lower limit of Tetration T A073230 [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...]
0.20787957635076190854695561983497877 i^i  i^i  e^ \frac{-\pi}{2} e^(-pi/2) T A049006 [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...]
0.28016949902386913303643649123067200 Bernstein constant \beta \frac {1}{2\sqrt {\pi}} T A073001 [0;3,1,1,3,9,6,3,1,3,13,1,16,3,3,4,…]
0.28878809508660242127889972192923078 Flajolet and Richmond Q  \prod_{n=1}^{\infty} \left(1 - \frac{1}{2^n}\right) = \left(1{-}\frac{1}{2^1}\right) \left(1{-}\frac{1}{2^2} \right)\left(1{-}\frac{1}{2^3} \right) ... prod[n=1 to ∞]{1-1/2^n} A048651
0.31830988618379067153776752674502872 Inverse of Pi, Ramanujan \frac{1}{\pi}  \frac{2\sqrt{2}}{9801} \sum^\infty_{n=0} \frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}} T A049541 [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,...]
0.47494937998792065033250463632798297 Weierstrass constant W_{_{WE}}  \frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4*2^{3/4} {(\frac {1}{4}!)^2}} (E^(Pi/8) Sqrt[Pi])/(4 2^(3/4) (1/4)!^2) T A094692 [0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,...]
0.56714329040978387299996866221035555 Omega constant \Omega W(1)=\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!} = 1 {-} 1 {+} \frac{3}{2} {-} \frac{8}{3} {+} \frac{125}{24} - ... sum[n=1 to ∞]{(-n)^(n-1)/n!} T A030178 [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,...]
0.57721566490153286060651209008240243 Euler's number \gamma  -\psi(1) = \sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} sum[n=1 to ∞]|sum[k=0 to ∞]{((-1)^k)/(2^n+k)} ? A001620 [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,...]
0.60459978807807261686469275254738524 Dirichlet serie  \frac{\pi}{3 \sqrt 3}  \sum_{n = 1}^\infty \frac{1}{n{2n \choose n}} =  1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{1}{7} - \frac{1}{8} + \cdots Sum[1/(n Binomial[2 n, n]), {n, 1, ∞}] T A073010 [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,...]
0.63661977236758134307553505349005745 2/Pi, François Viète \frac{2}{\pi}  \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots T A060294 [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]
0.66016181584686957392781211001455577 Twin prime constant C_{2} \prod_{p=3}^\infty \frac{p(p-2)}{(p-1)^2} prod[p=3 to ∞]{p(p-2)/(p-1)^2 A005597 [0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...]
0.66274341934918158097474209710925290 Laplace Limit constant \lambda A033259 [0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...]
0.69314718055994530941723212145817657 Logarithm de 2 Ln(2) \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots Sum[n=1 to ∞]{(-1)^(n+1)/n} T A002162 [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,...]
0.78343051071213440705926438652697546 Sophomore's Dream 1 J.Bernoulli I_{1} \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^n} = 1 - \frac{1}{2^2} + \frac{1}{3^3} - \frac{1}{4^4} + \frac{1}{5^5} + ... Sum[ -(-1)^n /n^n] T A083648 [0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,...]
0.78539816339744830961566084581987572 Dirichlet beta(1) \beta(1) \frac{\pi}{4} = \sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots Sum[n=0 to ∞]{(-1)^n/(2n+1)} T A003881 [0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,...]
0.82246703342411321823620758332301259 Traveling Salesman Nielsen-Ramanujan \frac{\zeta(2)}{2}  \frac{\pi^2}{12} = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \frac{1}{5^2} - ... Sum[n=1 to ∞]{((-1)^(k+1))/n^2} T A072691 [0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,...]
0.91596559417721901505460351493238411 Catalan constant C \sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^2} = \frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots Sum[n=0 to ∞]{(-1)^n/(2n+1)^2} I A006752 [0;1,10,1,8,1,88,4,1,1,7,22,1,2,...]
1.05946309435929526456182529494634170 Ratio of the distance between semi-tones \sqrt[12]{2} \sqrt[12]{2} 2^(1/12) I A010774 [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...]
1,.08232323371113819151600369654116790 Zeta(04) \zeta{4}  \frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ... Sum[n=1 to ∞]{1/n^4} T A013662 [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,...]
1.1319882487943 ... Constante de Viswanath C_{Vi} \lim_{n \to \infty}|a_n|^\frac{1}{n} A078416 [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...]
1.20205690315959428539973816151144999 Apéry constant \zeta(3) \sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots\,\! Sum[n=1 to ∞]{1/n^3} I A010774 [1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,...]
1.22541670246517764512909830336289053 Gamma(3/4) \Gamma(\tfrac34) \left(-1+\frac{3}{4}\right)! (-1+3/4)! T A068465 [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,...]
1.23370055013616982735431137498451889 Favard constant \tfrac34\zeta(2)  \frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+ ... sum[n=1 to ∞]{1/((2n-1)^2)} T A111003 [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...]
1.25992104989487316476721060727822835 Cube root of 2, constante Delian \sqrt[3]{2} \sqrt[3]{2} 2^(1/3) I A002580 [1;3,1,5,1,1,4,1,1,8,1,14,1,10,...]
1.29128599706266354040728259059560054 Sophomore's Dream 2 J.Bernoulli I_{2}  \sum_{n = 1}^\infty \frac{1}{n^n} =  1 + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4} + \frac{1}{5^5} + \frac{1}{6^6} + ... Sum[1/(n^n]), {n, 1, ∞}] A073009 [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,...]
1.32471795724474602596090885447809734 Plastic number \rho \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}}} I A060006 [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,...]
1.41421356237309504880168872420969808 Square root of 2, Pythagoras constant \sqrt{2} \prod_{n=1}^\infty 1+\frac{(-1)^{n+1}}{2n-1} = \left(1{+}\frac{1}{1}\right) \left(1{-}\frac{1}{3} \right)\left(1{+}\frac{1}{5} \right) ... prod[n=1 to ∞]{1+(-1)^(n+1)/(2n-1)} I A002193 [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
= [1;(2),...]
1.44466786100976613365833910859643022 Steiner number e^{\frac{1}{e}} e^{1/e} ... Upper Limit of Tetration A073229 [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
1.53960071783900203869106341467188655 Lieb's Square Ice constant W_{2D} \lim_{n \to \infty}(f(n))^{n^{-2}}=\left(\frac{4}{3}\right)^\frac{3}{2} (4/3)^(3/2) I A118273 [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...]
1.57079632679489661923132169163975144 Wallis product \pi/2  \prod_{n=1}^{\infty} (\frac{4n^2}{4n^2 - 1}) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots 
T A019669 [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...]
1.60669515241529176378330152319092458 Erdős–Borwein constant E_{\,B} \sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} + \frac{1}{3} + \frac{1}{7} + \frac{1}{15} + \cdots\,\! sum[n=1 to ∞]{1/(2^n-1)} I A065442 [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...]
1.61803398874989484820458633436563812 Phi, Golden ratio \varphi \frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}} (1+5^(1/2))/2 I A001622 [0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]
= [0;(1),...]
1.64493406684822643647241516664602519 Zeta(2) \zeta(\,2)  \frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots Sum[n=1 to ∞]{1/n^2} T A013661 [1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...]
1.66168794963359412129581892274995074 Somos' quadratic recurrence constant \sigma \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} = 1^{1/2} ; 2^{1/4} ; 3^{1/8} \cdots T A065481 [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]
1.73205080756887729352744634150587237 Theodorus constant \sqrt{3} \sqrt{3} 3^(1/2) I A002194 [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...]
= [1;(1,2),...]
1.75793275661800453270881963821813852 Kasner number R \sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}} A072449 [1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...]
1.77245385090551602729816748334114518 Carlson-Levin constant \Gamma(\tfrac12) \sqrt{\pi} = \left(-\frac{1}{2}\right)! sqrt (pi) T A002161 [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...]
2.29558714939263807403429804918949038 Universal parabolic constant  P_{\,2} \ln(1 + \sqrt2) + \sqrt2 ln(1+sqrt 2)+sqrt 2 T A103710 [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,...]
2.30277563773199464655961063373524797 Bronze Number \sigma_{\,Rr} \frac {3+\sqrt{13}}{2} = 1+ \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}} (3+sqrt 13)/2 I A098316 [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...]
= [3;(3),...]
2.37313822083125090564344595189447424 Lévy constant2 2\,ln\,\gamma \frac{\pi^2}{6ln(2)} Pi^(2)/(6*ln(2)) T A174606 [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...]
2.50662827463100050241576528481104525 square root of 2 pi \sqrt{2 \pi} \sqrt{2 \pi} = \lim_{n \to \infty} \frac {n! \; e^n}{n^n \sqrt{n}} sqrt (2*pi) T A019727 [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...]
2.66514414269022518865029724987313985 Gelfond-Schneider constant  G_{_{\,GS}} 2^{\sqrt{2}} 2^sqrt{2} T A007507 [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...]
2.68545200106530644530971483548179569 Khintchin constant  K_{\,0}  \prod_{n=1}^\infty \left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2} prod[n=1 to ∞]{(1+1/(n(n+2)))^((ln(n)/ln(2))} ? A002210 [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]
3.27582291872181115978768188245384386 Khinchin-Lévy constant \gamma  e^{\pi^2/(12\ln2)} e^(\pi^2/(12 ln(2)) A086702 [3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...]
3.35988566624317755317201130291892717 Reciprocal Fibonacci constant  \Psi \sum_{n=1}^{\infty} \frac{1}{F_n} = \frac{1}{1} +  \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \cdots A079586 [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
4.13273135412249293846939188429985264 Root of 2 e pi  \sqrt{2e \pi}  \sqrt{2e \pi} sqrt(2e pi) T A019633 [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...]
6.58088599101792097085154240388648649 Froda constant 2^{\,e} 2^e  2^e [6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...]
9.86960440108935861883449099987615114 Pi Squared \pi ^2 6 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{6}{1^2} + \frac{6}{2^2} + \frac{6}{3^2} + \frac{6}{4^2}+ \cdots 6 Sum[n=1 to ∞]{1/n^2} T A002388 [9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,...]
23.1406926327792690057290863679485474 Gelfond constant e^{\pi} \sum_{n=0}^\infty \frac{\pi^{n}}{n!} = \frac{\pi^{1}}{1} + \frac{\pi^{2}}{2!} + \frac{\pi^{3}}{3!} + \frac{\pi^{4}}{4!}+ \cdots Sum[n=0 to ∞]{(pi^n)/n!} T A039661 [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]

Books[change | edit source]

  • Finch, Steven (2003). Mathematical constants. Cambridge University Press. ISBN 0-521-81805-3.
  • Daniel Zwillinger (2012). Standard Mathematical Tables and Formulae. Imperial College Press.. ISBN 978-1-4398-3548-7.
  • Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics. Chapman & Hall/CRC. ISBN 1-58488-347-2.
  • Lloyd Kilford (2008). Modular Forms, a Classical and Computational Introduction. Imperial College Press.. ISBN 10 1-84816-213-8.

Online bibliography[change | edit source]

Some mathematical constants[change | edit source]

Here are some important mathematical constants:

Name Symbol Value Meaning
Pi, Archimedes' constant or Ludoph's number π ≈3.141592653589793 A transcendental number that is the ratio of the length of a circle's circumference to its diameter. It is also the area of the unit circle.
E, Napier's constant e ≈2.718281828459045 A transcendental number that is the base of natural logarithms, sometimes called the "natural number".
Golden ratio φ \frac{\sqrt{5}+1}{2} \approx 1.618 It is the value of a larger value divided by a smaller value if this is equal to the value of the sum of the values divided by the larger value.
Square root of 2, Pythagoras' constant \sqrt{2} \approx 1.414 An irrational number that is the length of the diagonal of a square with sides of length 1. This number can not be written as a fraction.

Other websites[change | edit source]